- First determine which input of \(f^{\prime}\) corresponds to \(x=2\). - The chain rule combines an outer derivative value with an inner derivative value. - All needed values are given; no explicit formulas for \(f\) or \(g\) are required.

1. By the chain rule, \(h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x)\). 2. At \(x=2\), the inner value is \(g(2)=5\), so \(f^{\prime}(g(2))=f^{\prime}(5)=-4\). 3. Therefore, \(h^{\prime}(2)=(-4)(3)=-12\).

\(-12\)

- Identify the outer power and the inner linear expression separately. - After differentiating the outer function, check whether its input also changes with \(x\). - Compare Dalia's expression with the full chain-rule structure.

1. The outer power contributes \(4(2x+1)^3\). 2. The inner function \(2x+1\) has derivative \(2\), so the chain rule requires an additional factor of \(2\). 3. Thus, \(p'(x)=8(2x+1)^3\).

The missing factor is \(2\), and \(p'(x)=8(2x+1)^3\).

- Which calculation happens first when you evaluate the expression? - Which operation is applied to the whole result?

1. The inner function is \(g(x)=3x+2\). 2. The outer function takes an input \(u\) and raises it to the fourth power: \(f(u)=u^4\). Thus \(h(x)=f(g(x))\).

Inner: \(g(x)=3x+2\). Outer: \(f(u)=u^4\).

- Which operation is performed last in each expression? - Try writing one expression as \(u(v(x))\).

1. In \(a(x)\), the outer function is \(u^3\) and the inner function is \(x^2+1\). The chain rule follows this structure directly. 2. In \(b(x)\), two factors are multiplied, so the product rule or expansion is more natural.

\(a(x)\). Inner: \(x^2+1\). Outer: \(u^3\). The expression \(b(x)\) is a product that can also be expanded.

- What stays unchanged inside the exponential when you differentiate it? - What factor comes from differentiating the exponent?

1. The derivative of the outer exponential is \(e^{5x}\). 2. The inner expression \(5x\) has derivative \(5\), so the chain rule gives \(5e^{5x}\).

\(5e^{5x}\)

- The input of cosine is not just \(x\). What is it? - How does the input \(2x\) change when \(x\) increases by a small amount?

1. The outer derivative is \(-\sin(2x)\). 2. The inner function \(2x\) has derivative \(2\), so the missing factor is \(2\).

The missing factor is \(2\).

- Identify the inner function and the outer power function. - Use the chain rule: outer derivative times inner derivative. - Include the constant factor \(\frac{1}{4}\) when differentiating the outer function.

1. Let the inner function be \(u(z)=2z^3+5\). Then \(u'(z)=6z^2\). 2. The outer function is \(\frac{1}{4}u^4\), whose derivative with respect to \(u\) is \(u^3\). 3. Apply the chain rule: \(h'(z)=(2z^3+5)^3(6z^2)\) \(=6z^2(2z^3+5)^3\).

\(h'(z)=6z^2(2z^3+5)^3\)

- Differentiate the two terms separately. - The sine term is a composite function. - What is the derivative of the expression inside the sine? - Multiply by the inner derivative when applying the chain rule.

1. Differentiate the two terms separately. 2. For the sine term, the inner function is \(u(t)=4t^2-\pi\), with \(u'(t)=8t\). By the chain rule, \(\frac{d}{dt}[2\sin(4t^2-\pi)]=2\cos(4t^2-\pi)(8t)\) \(=16t\cos(4t^2-\pi)\). 3. Also, \(\frac{d}{dt}(t^2)=2t\). Therefore, \(k'(t)=16t\cos(4t^2-\pi)+2t\).

\(k'(t)=16t\cos(4t^2-\pi)+2t\)

- Count the layers of composition from the outside inward. - Each layer should contribute a factor to the derivative. - What is the innermost function, and what is its derivative? - Rewriting the function with parentheses can make the layers clearer.

1. The function has three layers: the square, the sine function, and the inner linear function \(5x\). 2. Kofi differentiated the square and the sine but did not multiply by the derivative of \(5x\). 3. Applying the chain rule through all layers gives \(f'(x)=2\sin(5x)\cos(5x)\cdot5=10\sin(5x)\cos(5x)\).

The derivative of the innermost function \(5x\) was omitted. The correct derivative is \(f'(x)=10\sin(5x)\cos(5x)\).

- Identify the inner and outer functions. - Use the chain rule for each composite function. - Multiply by the derivative of the inner function. - Find the general derivative before substituting the given input.

1. For a), apply the chain rule: \(f'(x)=4(2x-5)^3\cdot 2=8(2x-5)^3\). Therefore, \(f'(3)=8 \cdot (6-5)^3=8\). 2. For b), \(g'(t)=\frac{1}{2\sqrt{t^2+7}}(2t)=\frac{t}{\sqrt{t^2+7}}\). Therefore, \(g'(3)=\frac{3}{\sqrt{16}}=\frac{3}{4}\).

a) \(f'(3)=8\) b) \(g'(3)=\frac{3}{4}\)

- Recall the derivative of an exponential function and the basic trigonometric functions. - Rewrite \(\sin^2(x)\) as \((\sin(x))^2\) to see the composition. - You may simplify \(2\sin(x)\cos(x)\) using a double-angle identity.

1. For a), apply the chain rule: \(f'(x)=e^{x^2-1}(2x)\). Therefore, \(f'(1)=e^0 \cdot 2=2\). 2. For b), write \(g(x)=(\sin(x))^2\). Then \(g'(x)=2\sin(x)\cos(x)\). At \(x=\frac{\pi}{4}\), \(g'\left(\frac{\pi}{4}\right)=2 \cdot \left(\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{\sqrt{2}}{2}\right)=1\).

a) \(f'(1)=2\) b) \(g'\left(\frac{\pi}{4}\right)=1\)

- The operation performed last is usually the outer function. - Identify the expression inside the outer power. - When differentiating the outer function, temporarily keep the inner input unchanged.

1. Choose \(u(x)=x^5\) and \(v(x)=2x^3+7\). 2. Their derivatives are \(u'(x)=5x^4\) and \(v'(x)=6x^2\). 3. Substitute \(v(x)\) into \(u'\): \(u'(v(x))=5(2x^3+7)^4\). 4. By the chain rule, \(f'(x)=u'(v(x))v'(x)\) \(=5(2x^3+7)^4(6x^2)\) \(=30x^2(2x^3+7)^4\).

a) \(u(x)=x^5\); \(v(x)=2x^3+7\) b) \(u'(x)=5x^4\); \(v'(x)=6x^2\) c) \(u'(v(x))=5(2x^3+7)^4\); \(f'(x)=30x^2(2x^3+7)^4\)

- Identify the inner and outer functions. - Use the rule “outer derivative times inner derivative.” - Differentiate the power while keeping the inner expression unchanged. - What is the derivative of the expression inside the parentheses?

1. Let the inner function be \(v(x)=2x^3-4\), so \(v'(x)=6x^2\). 2. The derivative of the outer function \(\frac{1}{2}v^5\) is \(\frac{5}{2}v^4\). 3. Apply the chain rule: \(f'(x)=\frac{5}{2}(2x^3-4)^4(6x^2)\) \(=15x^2(2x^3-4)^4\).

\(f'(x)=15x^2(2x^3-4)^4\)

- Identify which function is applied first and which is applied second. - Recall the chain rule for a composition. - Differentiate the outer and inner functions separately. - Multiply the outer derivative evaluated at the inner function by the inner derivative.

1. Let the outer function be \(u(v)=e^v\) and the inner function be \(v(x)=\sin(x)\). 2. Their derivatives are \(u'(v)=e^v\) and \(v'(x)=\cos(x)\). 3. By the chain rule, \(f'(x)=u'(v(x))v'(x)\) \(=e^{\sin(x)}\cos(x)\), which is the stated expression.

With \(u(v)=e^v\) and \(v(x)=\sin(x)\), the chain rule gives \(f'(x)=e^{\sin(x)}\cos(x)\).

- Identify the inner and outer functions. - Recall the derivatives of the natural logarithm and cosine. - Pay attention to the sign when differentiating the inner function. - Use a trigonometric identity to simplify the quotient.

1. Let the outer function be \(u(v)=\ln(v)\) and the inner function be \(v(x)=\cos(x)\). 2. Their derivatives are \(u'(v)=\frac{1}{v}\) and \(v'(x)=-\sin(x)\). 3. By the chain rule, \(g'(x)=u'(v(x))v'(x)\) \(=\frac{1}{\cos(x)}(-\sin(x))\) \(=-\frac{\sin(x)}{\cos(x)}=-\tan(x)\).

The chain rule gives \(g'(x)=\frac{1}{\cos(x)}(-\sin(x))=-\tan(x)\).

- Identify the operation performed last; it is usually the outer function. - Look for the expression inside the parentheses or under the radical. - Use the chain rule: derivative of the outer function times derivative of the inner function. - Rewriting a square root as a fractional power may help.

1. For a), choose \(u(t)=t^6\) and \(v(x)=2x^2-1\). Then \(u'(t)=6t^5\) and \(v'(x)=4x\). By the chain rule, \(f'(x)=6(2x^2-1)^5(4x)=24x(2x^2-1)^5\). 2. For b), choose \(u(t)=4\sqrt{t}\) and \(v(x)=x^3+2\). Then \(u'(t)=\frac{2}{\sqrt{t}}\) and \(v'(x)=3x^2\). By the chain rule, \(f'(x)=\frac{2}{\sqrt{x^3+2}}(3x^2)=\frac{6x^2}{\sqrt{x^3+2}}\). This derivative is defined for \(x>-\sqrt[3]{2}\).

a) \(u(t)=t^6\), \(v(x)=2x^2-1\); \(f'(x)=24x(2x^2-1)^5\) b) \(u(t)=4\sqrt{t}\), \(v(x)=x^3+2\); \(f'(x)=\frac{6x^2}{\sqrt{x^3+2}}\) for \(x>-\sqrt[3]{2}\)

- Identify the expression used as the input to the trigonometric function or power. - In a), keep the entire derivative of the inner function together. - Ask what operation is performed first and what operation is performed afterward. - Remember the negative sign in the derivative of cosine.

1. For a), choose \(u(t)=\cos(t)\) and \(v(x)=2x^4-x\). Since \(u'(t)=-\sin(t)\) and \(v'(x)=8x^3-1\), the chain rule gives \(f'(x)=-\sin(2x^4-x)(8x^3-1)\) \(=(1-8x^3)\sin(2x^4-x)\). 2. For b), choose \(u(t)=t^2\) and \(v(x)=\sin(x)-5\). Since \(u'(t)=2t\) and \(v'(x)=\cos(x)\), the chain rule gives \(f'(x)=2(\sin(x)-5)\cos(x)\).

a) \(u(t)=\cos(t)\), \(v(x)=2x^4-x\); \(f'(x)=(1-8x^3)\sin(2x^4-x)\) b) \(u(t)=t^2\), \(v(x)=\sin(x)-5\); \(f'(x)=2(\sin(x)-5)\cos(x)\)

- Identify the expression inside each exponent. - Differentiate the outer exponential function, then multiply by the derivative of the exponent. - Keep constant factors. - Rewrite a reciprocal exponential using a negative exponent.

1. For \(f(x)=e^{4x-2}\), the inner derivative is \(4\), so \(f'(x)=4e^{4x-2}\). 2. For \(g(x)=5e^{-0.2x}\), multiply by the inner derivative \(-0.2\): \(g'(x)=5\cdot(-0.2)e^{-0.2x}=-e^{-0.2x}\). 3. For \(h(x)=e^{x^2+3}\), the inner derivative is \(2x\), so \(h'(x)=2xe^{x^2+3}\). 4. Rewrite \(k(x)=e^{-2x}\). Then \(k'(x)=-2e^{-2x}=-\frac{2}{e^{2x}}\).

a) \(f'(x)=4e^{4x-2}\) b) \(g'(x)=-e^{-0.2x}\) c) \(h'(x)=2xe^{x^2+3}\) d) \(k'(x)=-2e^{-2x}=-\frac{2}{e^{2x}}\)

- Identify and differentiate the expression in each exponent. - Multiply the outer derivative by the inner derivative. - Rewrite a reciprocal exponential using a negative exponent. - Track negative signs carefully.

1. For \(f(x)=0.2e^{5-x^2}\), the inner derivative is \(-2x\). Thus, \(f'(x)=0.2\cdot(-2x)e^{5-x^2}=-0.4xe^{5-x^2}\). 2. For \(f(x)=e^{\frac{1}{2}x^2-x}\), the inner derivative is \(x-1\). Thus, \(f'(x)=(x-1)e^{\frac{1}{2}x^2-x}\). 3. Rewrite \(f(x)=-3e^{-4x-1}\). Then \(f'(x)=12e^{-4x-1}=\frac{12}{e^{4x+1}}\). 4. For \(f(x)=4e^{1-2x^3}\), the inner derivative is \(-6x^2\). Thus, \(f'(x)=-24x^2e^{1-2x^3}\).

a) \(f'(x)=-0.4xe^{5-x^2}\) b) \(f'(x)=(x-1)e^{\frac{1}{2}x^2-x}\) c) \(f'(x)=12e^{-4x-1}=\frac{12}{e^{4x+1}}\) d) \(f'(x)=-24x^2e^{1-2x^3}\)

- Identify the inner and outer functions in each term. - Multiply by the derivative of the inner function when applying the chain rule. - Remember the negative sign in the derivative of cosine.

1. For \(f(x)=\sin(4x)\), differentiate the outer sine function and multiply by the derivative of \(4x\): \(f'(x)=\cos(4x)\cdot 4=4\cos(4x)\). 2. For \(f(x)=5\cos(2x-1)\), \(f'(x)=5[-\sin(2x-1)]\cdot 2=-10\sin(2x-1)\). 3. Differentiate each term: \(\frac{d}{dx}\sin(0.5x)=0.5\cos(0.5x)\) and \(\frac{d}{dx}\cos(x^2)=-2x\sin(x^2)\). Therefore, \(f'(x)=0.5\cos(0.5x)-2x\sin(x^2)\).

1) \(f'(x)=4\cos(4x)\) 2) \(f'(x)=-10\sin(2x-1)\) 3) \(f'(x)=0.5\cos(0.5x)-2x\sin(x^2)\)

- First differentiate both functions using the chain rule. - Recall the derivatives of sine and cosine, including the sign. - Solve the Pythagorean identity for \(\sin^2(x)\). - Consider how adding a constant affects a derivative.

1. Apply the chain rule to \(u(x)=(\sin(x))^2\): \(u'(x)=2\sin(x)\cos(x)=\sin(2x)\). 2. Apply the chain rule to \(v(x)=-(\cos(x))^2\): \(v'(x)=-2\cos(x)(-\sin(x))=2\sin(x)\cos(x)=\sin(2x)\). 3. Thus, \(u'(x)=v'(x)\). 4. The Pythagorean identity gives \(\sin^2(x)=1-\cos^2(x)\), so \(u(x)=v(x)+1\). Functions that differ by a constant have the same derivative.

a) \(u'(x)=\sin(2x)\) and \(v'(x)=\sin(2x)\) b) The derivatives are equal because \(u(x)=v(x)+1\), so the functions differ only by a constant.

- Rewrite reciprocal exponentials using negative exponents. - Simplify the square root before differentiating. - Use the chain rule for each exponential function. - Treat \(k\) as a constant.

1. Rewrite the first function as \(f(x)=e^{4x}-3e^{-x}\). Then \(f'(x)=4e^{4x}+3e^{-x}\). 2. Since \(e^{6x}>0\), \(\sqrt{e^{6x}}=e^{3x}\). Therefore, \(f'(x)=3e^{3x}\). 3. Treat \(k\) as a constant. The inner derivative of \(kx^2\) is \(2kx\), so \(f'(x)=2kxe^{kx^2}\).

a) \(f'(x)=4e^{4x}+3e^{-x}\) b) \(f'(x)=3e^{3x}\) c) \(f'(x)=2kxe^{kx^2}\)

- Determine whether the radical or denominator can ever be zero. - Rewrite a radical denominator using a negative rational exponent. - Apply the chain rule to the composite power. - Multiply by the derivative of \(t^2+4\).

1. Since \(t^2+4\ge4\) for every real \(t\), the radical is positive and the denominator is never zero. Thus, \(D=\mathbb{R}\). 2. Rewrite \(g(t)=10(t^2+4)^{-\frac{1}{2}}\). 3. Apply the chain rule: \(g'(t)=10\left(-\frac{1}{2}\right)(t^2+4)^{-\frac{3}{2}}(2t)=-\frac{10t}{(t^2+4)^{\frac{3}{2}}}\).

\(D=\mathbb{R}\); \(g'(t)=-\frac{10t}{(t^2+4)^{\frac{3}{2}}}\)

- The slope at an input is the derivative value there. - Use the quotient property of logarithms before differentiating. - Include the inner derivative of \(2x+1\).

1. Rewrite the function: \(g(x)=\ln4-\ln(2x+1)\). 2. Differentiate using the chain rule: \(g'(x)=-\frac{2}{2x+1}\). 3. Evaluate at \(x=1.5\): \(g'(1.5)=-\frac{2}{2\cdot1.5+1}=-\frac{1}{2}\).

\(-\frac{1}{2}\)

- Identify the inner function inside the logarithm. - Differentiate the outer and inner functions. - Combine them using the chain rule. - Check whether the denominator can be zero when \(x>0\).

1. Apply the chain rule with inner function \(v(x)=x^2+4x\): \(f'(x)=\frac{v'(x)}{v(x)}=\frac{2x+4}{x^2+4x}\). 2. For every \(x>0\), the denominator \(x^2+4x=x(x+4)\) is positive. Therefore, the derivative is defined throughout the stated interval.

\(f'(x)=\frac{2x+4}{x^2+4x}\), with domain \((0, \infty)\)

- For a logarithm of a function, differentiate both the logarithm and its input. - Check the sign of each logarithm's argument before giving a domain. - A quadratic expression with a positive constant can behave differently from a linear one.

1. For a), use the logarithmic chain rule: \(f'(x)=\frac{-4}{1-4x}=\frac{4}{4x-1}\). The logarithm requires \(1-4x>0\), so \(x<\frac{1}{4}\). 2. For b), the inner derivative is \(10x\), so \(g'(x)=\frac{10x}{5x^2+1}\). The argument \(5x^2+1\) is always positive, so the derivative exists for every real \(x\).

a) \(f'(x)=\frac{4}{4x-1}\) for \(x<\frac{1}{4}\). b) \(g'(x)=\frac{10x}{5x^2+1}\) for all real \(x\).

- The natural logarithm equals zero when its input equals \(1\). - Solve the resulting linear equation. - Apply the chain rule to the logarithm.

1. Solve \(\ln(2x+e)=0\). Then \(2x+e=1\), so \(x=\frac{1-e}{2}\). 2. Apply the chain rule: \(g'(x)=\frac{2}{2x+e}\).

Zero: \(x=\frac{1-e}{2}\) \(g'(x)=\frac{2}{2x+e}\)

- Identify the inner and outer functions for the chain-rule method. - Use the square of a difference to expand the expression. - Expand the chain-rule result before comparing the two answers.

1. Using the chain rule, \(f'(x)=2(2x^3-5)(6x^2)\) \(=12x^2(2x^3-5)=24x^5-60x^2\). 2. Expanding first gives \(f(x)=4x^6-20x^3+25\). Differentiating term by term, \(f'(x)=24x^5-60x^2\). Both methods give the same derivative.

\(f'(x)=24x^5-60x^2\)

- Substitute the full inner expression into the outer function. - Differentiate the square of a linear expression with the chain rule. - Compare the result with the general form \(px+q\).

1. Substitute \(k(x)\) into \(g\): \(L(x)=m(ax+b)+c=(ma)x+(mb+c)\). Its slope is \(ma\). 2. The composition is \(h(x)=(mx+c)^2\). The chain rule gives \(h'(x)=2(mx+c)m=2m^2x+2mc\). 3. The derivative can be written \(px+q\) with \(p=2m^2\) and \(q=2mc\). If \(m=0\), it becomes the constant zero function.

a) \(L(x)=(ma)x+(mb+c)\); slope \(ma\). b) \(h'(x)=2m^2x+2mc\) c) It has the form \(px+q\) with \(p=2m^2\), \(q=2mc\), including \(p=q=0\) when \(m=0\).

- Identify the inner and outer functions. - Apply the rule “outer derivative times inner derivative.” - Pay close attention to signs, especially for cosine and the decreasing linear expression under the square root.

1. For a), \(f'(x)=\cos(x^2+3)(2x)=2x\cos(x^2+3)\). 2. For b), \(f'(x)=\frac{1}{2\sqrt{10-2x}}\cdot(-2)=-\frac{1}{\sqrt{10-2x}}\). This derivative is defined for \(x<5\). 3. For c), \(f'(x)=0.5[-\sin(4x)]\cdot 4=-2\sin(4x)\).

a) \(f'(x)=2x\cos(x^2+3)\) b) \(f'(x)=-\frac{1}{\sqrt{10-2x}}\) for \(x<5\) c) \(f'(x)=-2\sin(4x)\)

- The input of a logarithm must be positive. - A natural logarithm is zero when its input is one. - When differentiating the logarithm, do not forget that its input also changes with \(x\).

1. The logarithm needs \(6-2x>0\), so the domain is \(x<3\). 2. At the x-intercept, \(\ln(6-2x)=0\), so \(6-2x=1\), giving \(x=\frac{5}{2}\). 3. The chain rule gives \(f'(x)=\frac{-2}{6-2x}=\frac{1}{x-3}\), for \(x<3\).

a) \((-\infty,3)\) b) \(\left(\frac{5}{2},0\right)\) c) \(f'(x)=\frac{1}{x-3}\) for \(x<3\).

- To find a zero, isolate the exponential expression. - Use the natural logarithm to solve for its exponent. - Keep the exponent unchanged when differentiating the exponential factor.

1. Set \(h(x)=0\): \(e^{x-1}=3\). 2. Take natural logarithms: \(x-1=\ln3\), so the x-intercept is \((1+\ln3,0)\). 3. Differentiate using the chain rule: \(h'(x)=-e^{x-1}\).

X-intercept: \((1+\ln(3),0)\). \(h'(x)=-e^{x-1}\) for all real \(x\).

- Think of the logarithm as acting on one complete expression. - How do you differentiate a natural logarithm with a variable input? - What factor comes from the expression inside the logarithm?

1. The outer function is the natural logarithm; the inner expression is \(1+4x^2\). 2. The inner derivative is \(8x\). 3. The chain rule gives \(f'(x)=\frac{8x}{1+4x^2}\).

The inner derivative is \(8x\), so \(f'(x)=\frac{1}{1+4x^2}\cdot8x=\frac{8x}{1+4x^2}\).

- Which values of \(x\) make the expression inside the square root nonnegative? - Treat the square root as the outer function and the quadratic as its input. - At which inputs would the derivative formula have a zero denominator?

1. Factor the expression under the root: \(x^2+3x=x(x+3)\). It is nonnegative when \(x\leq-3\) or \(x\geq0\). The domain is \((-\infty,-3]\cup[0,\infty)\). 2. For \(x<-3\) or \(x>0\), differentiate the outer square root and the inner quadratic: \(f'(x)=\frac{2x+3}{2\sqrt{x^2+3x}}\). 3. At \(x=-3\) and \(x=0\), the denominator is zero. The slopes become unbounded when approaching these endpoints from within the domain, so there is no finite derivative there.

a) \((-\infty,-3]\cup[0,\infty)\) b) \(f'(x)=\frac{2x+3}{2\sqrt{x^2+3x}}\) for \(x<-3\) or \(x>0\). At \(-3\) and \(0\), the denominator approaches zero and the one-sided slopes become unbounded, so neither endpoint has a finite derivative.

- Write the chain rule before substituting any numbers. - Which input goes into \(f'\) when \(x=1\)? - Use the given slope to find the missing factor.

1. The chain rule gives \(h'(1)=f'(g(1))g'(1)\). 2. Substitute the known values: \(12=-3g'(1)\). 3. Divide by \(-3\) to get \(g'(1)=-4\).

\(g'(1)=-4\)

- The exponent is a function of x. What does the chain rule add when differentiating the exponential? - Compare the function's values at opposite x-values. - Use each point and its derivative to find the matching tangent line.

1. Using the chain rule, \(\phi'(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}(-x)=-x\phi(x)\). 2. At both given inputs, \(\phi(-1)=\phi(1)=c\), where \(c=\frac{1}{\sqrt{2\pi e}}\). 3. The slopes are \(\phi'(-1)=c\) and \(\phi'(1)=-c\). The tangent lines are \(y=c(x+2)\) and \(y=c(2-x)\), respectively. 4. The function values agree at opposite inputs because the exponent contains \(x^2\). The factor \(-x\) in \(\phi'(x)\) gives opposite slope signs.

a) \(\phi'(x)=-\frac{x}{\sqrt{2\pi}}e^{-x^2/2}\) b) At \(x=-1\): \(y=\frac{x+2}{\sqrt{2\pi e}}\). At \(x=1\): \(y=\frac{2-x}{\sqrt{2\pi e}}\). The slopes are opposites because \(\phi'(-x)=-\phi'(x)\).

- Use the chain rule when differentiating \(\sin(2x)\). - Evaluate both the function and derivative at the given input. - Use point-slope form for the tangent line. - For a negative slope, choose the inclination angle in the second quadrant.

1. Find the point of tangency: \(f\left(\frac{\pi}{3}\right)=2\sin\left(\frac{2\pi}{3}\right)+\cos\left(\frac{\pi}{3}\right)=\sqrt{3}+\frac{1}{2}\). 2. Differentiate using the chain rule: \(f'(x)=4\cos(2x)-\sin x\). The tangent slope is \(m=f'\left(\frac{\pi}{3}\right)=-2-\frac{\sqrt{3}}{2}\). 3. Use point-slope form: \(y-\left(\sqrt{3}+\frac{1}{2}\right)=\left(-2-\frac{\sqrt{3}}{2}\right)\left(x-\frac{\pi}{3}\right)\). Equivalently, \(y=\left(-2-\frac{\sqrt{3}}{2}\right)x+\frac{2\pi}{3}+\frac{\pi\sqrt{3}}{6}+\sqrt{3}+\frac{1}{2}\). 4. The angle of inclination \(\alpha\in[0^\circ, 180^\circ)\) satisfies \(\tan\alpha=m\). Since the slope is negative, \(\alpha\approx 109.23^\circ\).

Tangent line: \(y-\left(\sqrt{3}+\frac{1}{2}\right)=\left(-2-\frac{\sqrt{3}}{2}\right)\left(x-\frac{\pi}{3}\right)\) Angle of inclination: \(\alpha\approx 109.23^\circ\)

- The chain rule requires multiplying by the derivative of the inner function. - Differentiate the expression inside the parentheses separately. - Evaluate the derivative at the specified \(x\)-value to find the slope. - Track the sign when raising a negative number to an odd power.

1. Mirela differentiated the outer fourth power but omitted the derivative of the inner function. 2. The inner function is \(u(x)=2x^3-5\), with \(u'(x)=6x^2\). 3. By the chain rule, \(g'(x)=4(2x^3-5)^3(6x^2)=24x^2(2x^3-5)^3\). 4. Evaluate at \(x=1\): \(g'(1)=24\cdot1^2\cdot(2-5)^3=24\cdot(-3)^3=-648\).

The inner derivative was omitted. The correct derivative is \(g'(x)=24x^2(2x^3-5)^3\), and the slope at \(x=1\) is \(-648\).

- The chain rule has the form “outer derivative times inner derivative.” - Keep the original inner expression as the input of the differentiated outer function. - For an exponential function, check whether the original exponent has been preserved. - Look for factors that should have appeared from differentiating the inner function.

1. In a), the inner derivative is missing. Since \(\frac{d}{dx}(x^5-2)=5x^4\), \(f'(x)=4(x^5-2)^3(5x^4)=20x^4(x^5-2)^3\). 2. In b), the derivative of the inner function was incorrectly substituted into the sine argument. The original inner expression remains inside the outer derivative: \(g'(x)=-\sin(3x^2)(6x)=-6x\sin(3x^2)\). 3. In c), the exponent was incorrectly changed. For \(e^{u(x)}\), the exponential factor remains \(e^{u(x)}\): \(h'(x)=e^{4x+1}(4)=4e^{4x+1}\).

a) The inner derivative was omitted. \(f'(x)=20x^4(x^5-2)^3\). b) The argument of the sine function was changed incorrectly. \(g'(x)=-6x\sin(3x^2)\). c) The exponent was changed incorrectly. \(h'(x)=4e^{4x+1}\).

- Recall the derivative of the natural exponential function. - What sign appears when cosine is differentiated? - A slope at a point is found by evaluating the derivative there. - Use the unit-circle values of sine and cosine at \(\frac{\pi}{2}\).

1. Choose \(u(x)=e^x\) and \(v(x)=\cos(x)\). 2. Then \(u'(x)=e^x\), \(v'(x)=-\sin(x)\), and \(u'(v(x))=e^{\cos(x)}\). 3. By the chain rule, \(f'(x)=u'(v(x))v'(x)\) \(=-\sin(x)e^{\cos(x)}\). 4. At \(x=\frac{\pi}{2}\), \(f'\left(\frac{\pi}{2}\right)=-\sin\left(\frac{\pi}{2}\right)e^{\cos(\frac{\pi}{2})}\) \(=-1\cdot e^0=-1\).

a) \(u(x)=e^x\); \(v(x)=\cos(x)\) b) \(u'(x)=e^x\); \(v'(x)=-\sin(x)\); \(u'(v(x))=e^{\cos(x)}\) c) \(f'(x)=-\sin(x)e^{\cos(x)}\) d) \(f'\left(\frac{\pi}{2}\right)=-1\)

- Differentiate the two terms separately. - Rewrite the rational term using a negative exponent. - Identify the inner function in the cosine term. - Watch the negative sign when differentiating cosine. - Apply the chain rule to both terms.

1. Differentiate the first term using the chain rule: \(\frac{d}{dt}[3\cos(2t^2+5)]=-3\sin(2t^2+5)(4t)\) \(=-12t\sin(2t^2+5)\). 2. Rewrite the second term as \(-4(3t-2)^{-1}\). Then \(\frac{d}{dt}[-4(3t-2)^{-1}]=4(3t-2)^{-2}\cdot 3\) \(=\frac{12}{(3t-2)^2}\). 3. Combine the results: \(g'(t)=-12t\sin(2t^2+5)+\frac{12}{(3t-2)^2}\).

\(g'(t)=-12t\sin(2t^2+5)+\frac{12}{(3t-2)^2}\), for \(t\ne\frac{2}{3}\)

- Distinguish the variable from the parameters in each function. - Identify the inner and outer functions. - Watch the sign when differentiating cosine. - Make the power structure explicit before applying the chain rule.

1. For a), apply the chain rule: \(f_k'(x)=4(kx^2-5)^3(2kx)\) \(=8kx(kx^2-5)^3\). 2. For b), \(g_a'(t)=\cos(at-a)(a)\) \(=a\cos(at-a)\). 3. For c), write \(h_b(x)=(\cos(bx))^2\). Applying the chain rule through both layers gives \(h_b'(x)=2\cos(bx)[-b\sin(bx)]\) \(=-2b\sin(bx)\cos(bx)\).

a) \(f_k'(x)=8kx(kx^2-5)^3\) b) \(g_a'(t)=a\cos(at-a)\) c) \(h_b'(x)=-2b\sin(bx)\cos(bx)\)

- Identify the variable separately in each part. - Treat the other letters as constants when finding the inner derivative. - For the cosine expression, a square affects the input before differentiation.

1. For a), the inner derivative is \(3cz^2\), so \(f_c'(z)=3cz^2\cos(cz^3)\). 2. For b), the inner expression is \(a^2x^2\), with derivative \(2a^2x\). Thus \(g_a'(x)=-2a^2x\sin(a^2x^2)\). 3. For c), \(a\) and \(k\) are constants and the inner derivative is \(a\), so \(h_{a,k}'(t)=a\cos(at+k^2)\).

a) \(f_c'(z)=3cz^2\cos(cz^3)\) b) \(g_a'(x)=-2a^2x\sin(a^2x^2)\) c) \(h_{a,k}'(t)=a\cos(at+k^2)\)

- Use the chain rule for a power of a linear expression. - A horizontal tangent has derivative \(0\). - Convert the given angle to a slope. - Find the y-coordinate after solving for the x-coordinate.

1. Differentiate using the chain rule: \(f'(x)=\frac{1}{2}\cdot 4(0.5x+1)^3\cdot 0.5=(0.5x+1)^3\). 2. At \(x=2\), \(f'(2)=(2)^3=8\). 3. A horizontal tangent has slope \(0\). Solve \((0.5x+1)^3=0\), giving \(x=-2\). Since \(f(-2)=0\), the point is \((-2,0)\). 4. An angle of \(45^\circ\) corresponds to slope \(1\). Solve \((0.5x+1)^3=1\), giving \(x=0\). 5. Since \(f(0)=\frac{1}{2}\), the point is \((0,\frac{1}{2})\).

a) \(8\) b) Yes; the point is \((-2,0)\). c) \((0,\frac{1}{2})\)

- Use the chain rule and track the sign of the inner derivative. - A horizontal tangent has slope \(0\). - Convert \(135^\circ\) to a slope. - A squared equation may have two solutions.

1. Differentiate using the chain rule: \(f'(x)=-(2-x)^2\). 2. At \(x=1\), \(f(1)=\frac{1}{3}\) and \(f'(1)=-1\). The tangent line is \(y-\frac{1}{3}=-(x-1)\), or \(y=-x+\frac{4}{3}\). 3. A horizontal tangent has slope \(0\). Solve \(-(2-x)^2=0\), giving \(x=2\). Since \(f(2)=0\), the point is \((2,0)\). 4. An angle of \(135^\circ\) corresponds to slope \(-1\). Solve \(-(2-x)^2=-1\), so \((2-x)^2=1\). Thus, \(x=1\) or \(x=3\). 5. The corresponding points are \((1,\frac{1}{3})\) and \((3,-\frac{1}{3})\).

a) \(y=-x+\frac{4}{3}\) b) \((2,0)\) c) \((1,\frac{1}{3})\) and \((3,-\frac{1}{3})\)

- For each part, identify the inner and outer functions. - Use the chain rule: differentiate the outer function, then multiply by the derivative of the inner function. - In b), remember how constants behave under differentiation. - Rewriting a reciprocal or square root as a power may help.

1. For a), the inner function is \(e^x\), whose derivative is \(e^x\). Therefore, \(g'(x)=e^x f'(e^x)\). 2. For b), apply the chain rule to the outer function \(u^3\) and the inner function \(f(x)+k\): \(g'(x)=3(f(x)+k)^2f'(x)\). 3. For c), the inner function is \(x^{-1}\), whose derivative is \(-x^{-2}\). Therefore, \(g'(x)=-\frac{1}{x^2}f'\left(\frac{1}{x}\right)\). 4. For d), differentiate the outer function \(\sqrt{u}\) and multiply by the derivative of the inner function: \(g'(x)=\frac{f'(x)}{2\sqrt{f(x)}}\).

a) \(g'(x)=e^x f'(e^x)\) b) \(g'(x)=3(f(x)+k)^2f'(x)\) c) \(g'(x)=-\frac{1}{x^2}f'\left(\frac{1}{x}\right)\) d) \(g'(x)=\frac{f'(x)}{2\sqrt{f(x)}}\)

- Treat \(a\), \(b\), \(c\), and \(d\) as constants. - Identify the inner and outer functions in each composition. - Apply the power rule together with the chain rule. - For \(f(f(x))\), distinguish the derivative of the outer copy of \(f\) from the derivative of the inner copy.

1. For a), the derivative of the inner function \(ax^n+b\) is \(anx^{n-1}\). Thus, \(g'(x)=anx^{n-1}f'(ax^n+b)\). 2. For b), differentiate the outer function \(\frac{1}{n}u^n\), then multiply by \(f'(x)\): \(g'(x)=(f(x))^{n-1}f'(x)\). 3. For c), the outer copy of \(f\) is evaluated at \(f(x)\), while the inner derivative is \(f'(x)\). Therefore, \(g'(x)=f'(f(x))f'(x)\). 4. For d), the constant \(d\) differentiates to zero, the factor \(a\) remains, and the inner linear function has derivative \(b\). Therefore, \(g'(x)=abf'(bx+c)\).

a) \(g'(x)=anx^{n-1}f'(ax^n+b)\) b) \(g'(x)=(f(x))^{n-1}f'(x)\) c) \(g'(x)=f'(f(x))f'(x)\) d) \(g'(x)=abf'(bx+c)\)

- What equation characterizes a function whose graph is symmetric about the y-axis? - Use the chain rule for a function of the form \(f(u(x))\). - What equation characterizes a function whose graph is symmetric about the origin? - When substituting \(-x\) into \(g'(x)\), track the sign of the linear factor carefully.

1. For a), \(g(-x)=f((-x)^2)=f(x^2)=g(x)\). Therefore, \(g\) is even, so its graph is symmetric about the y-axis. 2. For b), apply the chain rule: \(g'(x)=f'(x^2)(2x)=2x f'(x^2)\). 3. For c), \(g'(-x)=2(-x)f'((-x)^2)\) \(=-2x f'(x^2)=-g'(x)\). Therefore, \(g'\) is odd, so its graph is symmetric about the origin.

a) Since \(g(-x)=f((-x)^2)=f(x^2)=g(x)\), the graph of \(g\) is symmetric about the y-axis. b) \(g'(x)=2x f'(x^2)\) c) Since \(g'(-x)=-2x f'(x^2)=-g'(x)\), the graph of \(g'\) is symmetric about the origin.

- Identify the operation performed last; it gives the outermost function. - Work from the outside inward to identify all three layers. - Apply the chain rule once for each layer of the composition. - Multiply the derivative factors in outer-to-inner order.

1. One valid decomposition is \(u(t)=e^t\), \(v(t)=\cos(t)\), and \(w(x)=2x+1\). 2. Their derivatives are \(u'(t)=e^t\), \(v'(t)=-\sin(t)\), and \(w'(x)=2\). 3. Applying the chain rule to the three-function composition gives \(f'(x)=u'(v(w(x)))v'(w(x))w'(x)\). 4. Substitute the functions and derivatives: \(f'(x)=e^{\cos(2x+1)}(-\sin(2x+1))\cdot 2\) \(=-2\sin(2x+1)e^{\cos(2x+1)}\).

a) One valid choice is \(u(t)=e^t\), \(v(t)=\cos(t)\), and \(w(x)=2x+1\). b) \(f'(x)=-2\sin(2x+1)e^{\cos(2x+1)}\)

- Evaluate the function at \(\pi\). - Use the chain rule for \(\cos^2x\). - The derivative value is the tangent slope. - Use the negative reciprocal for the normal slope.

1. Evaluate the function: \(f(\pi)=\cos^2\pi-\sin\pi=1\). Thus, \(P=(\pi, 1)\). 2. Differentiate using the chain rule: \(f'(x)=2\cos x(-\sin x)-\cos x\) \(=-\sin(2x)-\cos x\). Therefore, \(f'(\pi)=1\). 3. The tangent line is \(y-1=x-\pi\), or \(y=x-\pi+1\). 4. The normal slope is \(-1\), so the normal line is \(y-1=-(x-\pi)\), or \(y=-x+\pi+1\).

Tangent line: \(y=x-\pi+1\) Normal line: \(y=-x+\pi+1\)

- Evaluate the function to find the point of tangency. - Differentiate using the chain rule. - Use the derivative value as the tangent slope. - The normal slope is the negative reciprocal of the tangent slope.

1. Find the point: \(f(0)=e^0+e^0=2\), so the point is \((0, 2)\). 2. Differentiate using the chain rule: \(f'(x)=2e^{2x}+e^x\). Thus, the tangent slope is \(m_t=f'(0)=2+1=3\). The tangent line is \(t:y-2=3(x-0)\), or \(t:y=3x+2\). 3. The normal slope is the negative reciprocal: \(m_n=-\frac{1}{3}\). Therefore, \(n:y-2=-\frac{1}{3}(x-0)\), or \(n:y=-\frac{1}{3}x+2\).

\(t:y=3x+2\) \(n:y=-\frac{1}{3}x+2\)

- Translate the point and slope information into two equations. - Use the chain rule for \(e^{2x}\). - Solve the resulting system of two linear equations.

1. The point condition gives \(g_{r,s}(0)=r+s=1\). 2. Differentiate using the chain rule: \(g_{r,s}'(x)=2re^{2x}+se^x\). 3. The slope condition gives \(g_{r,s}'(0)=2r+s=5\). 4. Subtract \(r+s=1\) from \(2r+s=5\) to obtain \(r=4\). 5. Substitute into \(r+s=1\): \(s=-3\).

\(r=4\) and \(s=-3\)

- Use the chain rule for \(e^{ax}\). - A point on the x-axis has y-coordinate \(0\). - Write the tangent line in terms of \(a\). - Substitute the given x-intercept into the tangent equation.

1. Apply the chain rule: \(g_a'(x)=2ae^{ax}\). 2. At \(x=0\), \(g_a(0)=3\) and \(g_a'(0)=2a\). 3. The tangent line is \(y=2ax+3\). 4. It must pass through \((-1, 0)\), so \(0=-2a+3\). 5. Therefore, \(a=\frac{3}{2}\).

a) \(g_a'(x)=2ae^{ax}\) b) \(a=\frac{3}{2}\)

- Find both the function value and slope at \(x=1\). - Simplify the tangent equation before applying the conditions. - Positive slope determines the sign of \(a\). - Set the tangent equation equal to \(0\) to find its x-intercept. - Translate the x-intercept condition into an inequality for \(a\).

1. At \(x=1\), \(g_a(1)=a+2\). 2. Differentiate: \(g_a'(x)=ae^{x-1}\), so the tangent slope at \(x=1\) is \(a\). 3. The tangent line is \(y-(a+2)=a(x-1)\), which simplifies to \(y=ax+2\). 4. Positive slope requires \(a>0\). 5. The x-intercept is \(x=-\frac{2}{a}\). Require \(-\frac{2}{a}>-1\). 6. Since \(a>0\), multiplication by \(a\) preserves the inequality: \(-2>-a\), so \(a>2\).

a) \(y=ax+2\) b) \(a>2\)

- Identify the nested functions in the exponent. - Apply the chain rule to \(e^{u(x)}\). - The exponent is itself a composition, so apply the chain rule again. - Recall the derivative of \(\cos(ax)\).

1. Let \(u(x)=\cos(2x)\). Applying the chain rule to the exponential function gives \(g'(x)=\frac{1}{2}e^{\cos(2x)}u'(x)\). 2. Apply the chain rule again to the cosine function: \(u'(x)=-\sin(2x)\cdot 2=-2\sin(2x)\). 3. Substitute and simplify: \(g'(x)=\frac{1}{2}e^{\cos(2x)}(-2\sin(2x))\) \(=-\sin(2x)e^{\cos(2x)}\).

\(g'(x)=-\sin(2x)e^{\cos(2x)}\)

- Convert the point information into an equation. - Differentiate to convert the slope information into a second equation. - Two unknown parameters require two independent equations. - Substitute the repeated expression from one equation into the other.

1. The point condition gives \(ae^b=e^2\). 2. Apply the chain rule: \(f_{a,b}'(x)=2abxe^{bx^2}\). 3. The slope condition at \(x=1\) gives \(2abe^b=4e^2\). 4. Substitute \(ae^b=e^2\): \(2be^2=4e^2\), so \(b=2\). 5. Then \(ae^2=e^2\), so \(a=1\).

\(a=1\) and \(b=2\)

- The first derivative gives the slope. - What is the maximum value of the cosine function? - Express the maximum slope as a function of \(k\).

1. Apply the chain rule: \(f_k'(x)=k\cos(kx)\). 2. Since the maximum value of cosine is \(1\), the maximum value of the derivative is \(k\). Therefore, the maximum slope is \(k\). 3. Replacing \(k\) by \(2k\) changes the maximum slope from \(k\) to \(2k\). 4. Thus, doubling \(k\) doubles the maximum slope.

The maximum slope is \(k\). When \(k\) is doubled, the maximum slope doubles to \(2k\).

- View each squared trigonometric expression as a composition. - Apply the power rule before differentiating the trigonometric inner function. - Remember to differentiate the linear expressions \(3x\) and \(2x\). - Look for a product that matches the double-angle identity.

1. Write \(f(x)=(\sin(3x))^2\). Applying the chain rule twice gives \(f'(x)=2\sin(3x)\cos(3x)\cdot 3\). Since \(2\sin(3x)\cos(3x)=\sin(6x)\), \(f'(x)=3\sin(6x)\). 2. Differentiate each term of \(g\): \(g'(x)=\frac{3}{2}[2\cos(x)(-\sin(x))]-\frac{3}{4}[-\sin(2x)\cdot 2]\) \(=-3\sin(x)\cos(x)+\frac{3}{2}\sin(2x)\). Using \(2\sin(x)\cos(x)=\sin(2x)\), \(g'(x)=-\frac{3}{2}\sin(2x)+\frac{3}{2}\sin(2x)=0\).

1) \(f'(x)=3\sin(6x)\) 2) \(g'(x)=0\)

- Set the two function values equal to find their intersection. - A substitution involving \(e^x\) can turn the intersection equation into a quadratic equation. - Compare both the function values and derivative values at the intersection. - Parallel lines have equal slopes.

1. Set the functions equal: \(e^x=2-e^{-x}\). Multiplying by \(e^x\) gives \(e^{2x}-2e^x+1=0\). Let \(u=e^x\). Then \((u-1)^2=0\), so \(u=1\), \(x=0\), and the intersection point is \(S(0, 1)\). 2. Differentiate: \(f'(x)=e^x\) and \(g'(x)=e^{-x}\). Thus \(f'(0)=g'(0)=1\), so the graphs share the tangent line \(y=x+1\) at \(S\). 3. To compare the graphs, \(f(x)-g(x)=e^x+e^{-x}-2\geq0\), with equality only at \(x=0\). Therefore, the graph of \(f\) stays above the graph of \(g\), and the graphs touch without crossing at \(S\). 4. At the same x-value, parallel tangent lines require \(f'(x)=g'(x)\). Solving \(e^x=e^{-x}\) gives \(e^{2x}=1\), so \(x=0\) is the only solution.

a) \(S(0, 1)\) b) \(f'(0)=g'(0)=1\). The graphs share the tangent line \(y=x+1\), and \(f(x)\geq g(x)\) with equality only at \(x=0\), so they do not cross. c) No. At the same x-value, the tangent lines are parallel only when \(x=0\).

- For each exponential term, keep the original exponent when differentiating. - Check the derivative of the input to sine, including the minus sign. - In part c, there is a function inside a square inside an exponential.

1. For a), the chain rule on the exponential gives \(f'(x)=8(-0.5)e^{-0.5x}+10x^4=-4e^{-0.5x}+10x^4\). 2. For b), differentiate both compositions: \(g'(x)=e^{5x+2}+4\cos(-4x)=e^{5x+2}+4\cos(4x)\). 3. For c), the inner exponent is \((3x-4)^2\), with derivative \(6(3x-4)\). Therefore \(h'(x)=6(3x-4)e^{(3x-4)^2}\).

a) \(f'(x)=-4e^{-0.5x}+10x^4\) b) \(g'(x)=e^{5x+2}+4\cos(4x)\) c) \(h'(x)=6(3x-4)e^{(3x-4)^2}\)

- For a) and b), decide how the exponential and square-root factors are combined before differentiating. - Differentiate the exponent or the square-root input without losing the other factor. - A derivative formula for positive inputs does not settle what happens at the endpoint. What limit could you use there?

1. Both a) and b) are defined on \([0,\infty)\). For \(x>0\), the product and chain rules give \(f'(x)=2xe^{\sqrt{x}}+\frac{x^2e^{\sqrt{x}}}{2\sqrt{x}}=xe^{\sqrt{x}}\left(2+\frac{\sqrt{x}}{2}\right)\). 2. The right-hand difference quotient of a) at zero is \(he^{\sqrt{h}}\), which tends to \(0\). Thus the right-hand derivative at zero is \(0\), but a two-sided derivative there is not defined. 3. Write b) as \(g(x)=\sqrt{x}e^{-x^2}\). For \(x>0\), the product and chain rules give \(g'(x)=\frac{e^{-x^2}}{2\sqrt{x}}-2x\sqrt{x}e^{-x^2}=\frac{1-4x^2}{2\sqrt{x}e^{x^2}}\). At zero, the right-hand difference quotient \(e^{-h^2}/\sqrt{h}\) grows without bound, so no finite right-hand derivative exists. 4. For c), the quantity under the square root is positive for every real \(x\). The chain rule gives \(h'(x)=\frac{3e^{3x}}{2\sqrt{e^{3x}+1}}\) for every real \(x\).

a) \(f'(x)=xe^{\sqrt{x}}\left(2+\frac{\sqrt{x}}{2}\right)\) for \(x>0\); right-hand derivative at \(0\) is \(0\). b) \(g'(x)=\frac{1-4x^2}{2\sqrt{x}e^{x^2}}\) for \(x>0\); no finite right-hand derivative at \(0\). c) \(h'(x)=\frac{3e^{3x}}{2\sqrt{e^{3x}+1}}\) for all real \(x\).

- The derivative at a point gives the graph's slope there. - Rewrite \(\sqrt{e^x}\) as a single exponential expression before differentiating. - “Greater slope” means the larger derivative value. - Evaluate both derivatives at \(x=1\) and compare.

1. By the chain rule, \(f'(x)=e^{\sqrt{x}}\frac{1}{2\sqrt{x}}=\frac{e^{\sqrt{x}}}{2\sqrt{x}}\). 2. Since \(g(x)=e^{x/2}\), \(g'(x)=\frac{1}{2}e^{x/2}\). 3. At \(x=1\), \(f'(1)=\frac{e}{2}\) and \(g'(1)=\frac{\sqrt{e}}{2}\). Because \(e>\sqrt{e}\), \(f'(1)>g'(1)\).

\(f'(x)=\frac{e^{\sqrt{x}}}{2\sqrt{x}}\) and \(g'(x)=\frac{1}{2}e^{x/2}\). The graph of \(f\) has the greater slope at \(x=1\).

- Identify which function is the inner function in each composition. - Use the chain rule for a function of the form \(u(v(x))\). - Recall the derivative rules for logarithmic and power functions. - In part 2, the square applies to the entire logarithmic expression.

1. The first composition is \(h(x)=\ln(2x^2+5)\). By the chain rule, \(h'(x)=\frac{1}{2x^2+5}(4x)=\frac{4x}{2x^2+5}\). Since \(2x^2+5>0\) for every real \(x\), the domain of \(h\) is all real numbers. 2. The reverse composition is \(k(x)=2(\ln(x))^2+5\), which is defined for \(x>0\). Applying the chain rule gives \(k'(x)=4\ln(x)\left(\frac{1}{x}\right)=\frac{4\ln(x)}{x}\).

1) \(h(x)=\ln(2x^2+5)\), \(h'(x)=\frac{4x}{2x^2+5}\), and the domain of \(h\) is \((-\infty, \infty)\). 2) \(k(x)=2(\ln(x))^2+5\), \(k'(x)=\frac{4\ln(x)}{x}\), and the domain of \(k\) is \((0, \infty)\).

- Rewrite the expression as an outer power applied to an inner polynomial. - A horizontal tangent occurs where the derivative is zero. - Apply the power rule and then multiply by the derivative of the inner function. - A fraction equals zero when its numerator is zero and its denominator is nonzero.

1. One possible decomposition is \(u(t)=t^{-2}\) and \(v(x)=x^3-4\). 2. Since \(u'(t)=-2t^{-3}\) and \(v'(x)=3x^2\), the chain rule gives \(p'(x)=-2(x^3-4)^{-3}(3x^2)\) \(=-\frac{6x^2}{(x^3-4)^3}\). 3. A horizontal tangent occurs where \(p'(x)=0\). A rational expression is zero when its numerator is zero and its denominator is nonzero. Thus, \(-6x^2=0\), so \(x=0\). At \(x=0\), the denominator is nonzero, so this value is in the domain.

1) One possible choice is \(u(t)=t^{-2}\) and \(v(x)=x^3-4\). 2) \(p'(x)=-\frac{6x^2}{(x^3-4)^3}\) 3) \(x=0\)

- Rewrite the cube root as a rational power. - Identify the outer, middle, and inner functions. - Remember the derivative of \(\sin(2x)\), including the derivative of its argument. - A negative rational exponent can be rewritten as a radical in the denominator.

1. Rewrite the function as \(f(x)=(\sin(2x)+2)^{\frac{1}{3}}\). 2. Apply the chain rule to the outer power and then to \(\sin(2x)\): \(f'(x)=\frac{1}{3}(\sin(2x)+2)^{-\frac{2}{3}}\cdot2\cos(2x)\). 3. Rewrite with a radical: \(f'(x)=\frac{2\cos(2x)}{3\sqrt[3]{(\sin(2x)+2)^2}}\).

\(f'(x)=\frac{2\cos(2x)}{3\sqrt[3]{(\sin(2x)+2)^2}}\)

- Match each graph transformation to its parameter in the function formula. - A shift left changes the expression inside the radical to \(x+4\). - Differentiate the square-root function with the chain rule. - The derivative at \(x=5\) gives the tangent slope.

1. The transformations give \(a=3\), \(b=4\), and \(c=-2\), so \(f(x)=3\sqrt{x+4}-2\). 2. Differentiate: \(f'(x)=\frac{3}{2\sqrt{x+4}}\). 3. At \(x=5\), the slope is \(f'(5)=\frac{3}{2\sqrt{9}}=\frac{1}{2}\), and the point is \((5,7)\). 4. Using point-slope form, \(y-7=\frac{1}{2}(x-5)\), so \(y=\frac{1}{2}x+\frac{9}{2}\).

1. \(f(x)=3\sqrt{x+4}-2\) 2. \(y=\frac{1}{2}x+\frac{9}{2}\)

- Use radical properties to factor the expression before describing the transformations. - A factor outside the radical creates a vertical stretch. - The derivative gives the graph's slope. - Set the derivative equal to the target slope and solve.

1. Rewrite \(f(x)=\sqrt{4(x+3)}=2\sqrt{x+3}\). The graph is vertically stretched by a factor of \(2\) and shifted \(3\) units left. 2. Differentiate: \(f'(x)=\frac{1}{\sqrt{x+3}}\). 3. Set the derivative equal to \(1\): \(\frac{1}{\sqrt{x+3}}=1\). Thus, \(x+3=1\), so \(x_0=-2\).

1. \(f(x)=2\sqrt{x+3}\); vertical stretch by \(2\) and shift \(3\) units left 2. \(x_0=-2\)

- A square-root radicand must be nonnegative. - Determine where the downward-opening quadratic is nonnegative relative to its zeros. - Differentiating a square root places the radical in the denominator. - Division by zero is undefined.

1. The radicand must be nonnegative: \(12-x-x^2\ge0\). 2. Factor the corresponding quadratic: \(12-x-x^2=-(x+4)(x-3)\). It is nonnegative between its zeros, so \(D_f=[-4,3]\). 3. For interior points, \(f'(x)=\frac{-1-2x}{2\sqrt{12-x-x^2}}\). 4. This derivative is defined when the radicand is strictly positive, so \(D_{f'}=(-4,3)\). At \(x=-4\) and \(x=3\), the graph has vertical tangent behavior and no finite two-sided derivative.

a) \(D_f=[-4,3]\) b) \(D_{f'}=(-4,3)\)

- Require each square-root radicand to be nonnegative. - Set the function values equal to find an intersection. - After squaring, check the resulting solution in the original equation. - The derivative at the intersection gives the tangent slope. - Use point-slope form for the tangent line.

1. The radicands must be nonnegative, so \(D_f=[-4,\infty)\) and \(D_g=[0,\infty)\). 2. Set the functions equal: \(\sqrt{x+4}=4-\sqrt{x}\). Squaring and simplifying gives \(8\sqrt{x}=12\), so \(\sqrt{x}=\frac{3}{2}\) and \(x=\frac{9}{4}\). Then \(y=\frac{5}{2}\), so \(S=\left(\frac{9}{4},\frac{5}{2}\right)\). 3. Since \(f'(x)=\frac{1}{2\sqrt{x+4}}\), the tangent slope is \(f'\left(\frac{9}{4}\right)=\frac{1}{5}\). 4. Using point-slope form, \(y-\frac{5}{2}=\frac{1}{5}\left(x-\frac{9}{4}\right)\), so \(y=\frac{1}{5}x+\frac{41}{20}\).

a) \(D_f=[-4,\infty)\); \(D_g=[0,\infty)\) b) \(S=\left(\frac{9}{4},\frac{5}{2}\right)\) c) \(y=\frac{1}{5}x+\frac{41}{20}\)

- Set \(y=0\) for x-intercepts and \(x=0\) for y-intercepts. - Set the two function expressions equal to find shared points. - Apply the chain rule to each square-root function. - Two nonvertical lines are perpendicular when the product of their slopes is \(-1\).

1. a) For \(f\), \(f(0)=0\), so the only axis intercept is \((0,0)\). For \(g\), \(g(0)=\sqrt{6}\), giving \((0,\sqrt{6})\), and \(g(6)=0\), giving \((6,0)\). 2. b) Set the functions equal: \(\sqrt{2x}=\sqrt{6-x}\). Squaring gives \(2x=6-x\), so \(x=2\) and \(y=2\). The resulting linear equation has only one solution, and it checks in the original equation. 3. c) The derivatives are \(f'(x)=\frac{1}{\sqrt{2x}}\) and \(g'(x)=-\frac{1}{2\sqrt{6-x}}\). 4. At \(x=2\), the slopes are \(m_f=\frac{1}{2}\) and \(m_g=-\frac{1}{4}\). Their product is \(-\frac{1}{8}\ne-1\), so the tangent lines are not perpendicular.

a) For \(f\): \((0,0)\); for \(g\): \((0,\sqrt{6})\) and \((6,0)\) b) The only shared point is \((2,2)\). c) No. The tangent slopes have product \(-\frac{1}{8}\), not \(-1\).

- A square-root radicand must be nonnegative. - A tangent line requires both the point on the graph and the derivative there. - Use the chain rule to differentiate the square root. - A point on the y-axis has \(x=0\).

1. a) Require \(6-3x\ge0\), which gives \(x\le2\). Thus, \(D_f=(-\infty,2]\). 2. b) The point of tangency is \((-1,f(-1))=(-1,5)\). 3. Differentiate: \(f'(x)=-\frac{3}{2\sqrt{6-3x}}\), defined for \(x<2\). Therefore, \(f'(-1)=-\frac{1}{2}\). 4. Using point-slope form, \(y-5=-\frac{1}{2}(x+1)\), so \(t\) is \(y=-\frac{1}{2}x+\frac{9}{2}\). 5. c) Set \(x=0\). The y-intercept is \(\left(0,\frac{9}{2}\right)\).

a) \(D_f=(-\infty,2]\) b) \(t:y=-\frac{1}{2}x+\frac{9}{2}\) c) \(\left(0,\frac{9}{2}\right)\)

- Require the expression under the square root to be nonnegative. - Set the derivative equal to the specified tangent slope. - To solve a reciprocal square-root equation, isolate the radical before squaring. - A line is determined by a slope and one point.

1. a) Require \(2x+1\ge0\), so \(D_k=\left[-\frac{1}{2},\infty\right)\). 2. Differentiate: \(k'(x)=-\frac{1}{\sqrt{2x+1}}\). 3. Set \(k'(x_0)=-\frac{1}{5}\): \(\sqrt{2x_0+1}=5\), so \(2x_0+1=25\) and \(x_0=12\). 4. The point of tangency is \((12,k(12))=(12,-1)\). 5. Using point-slope form, \(y+1=-\frac{1}{5}(x-12)\), so \(y=-\frac{1}{5}x+\frac{7}{5}\).

a) \(D_k=\left[-\frac{1}{2},\infty\right)\) b) \(x_0=12\) c) \(y=-\frac{1}{5}x+\frac{7}{5}\)

- A square-root radicand must be nonnegative. - Apply the chain rule to the square-root function. - A tangent line needs the function value and derivative value at the given input. - Use point-slope form after finding the slope.

1. a) Require \(12-3x^2\ge0\), or \(x^2\le4\). Therefore, \(D_f=[-2,2]\). 2. b) By the chain rule, \(f'(x)=\frac{-6x}{2\sqrt{12-3x^2}}=-\frac{3x}{\sqrt{12-3x^2}}\). 3. c) At \(x=1\), \(f(1)=3\) and \(f'(1)=-1\). 4. Using point-slope form, \(y-3=-(x-1)\), so \(y=-x+4\).

a) \(D_f=[-2,2]\) b) \(f'(x)=-\frac{3x}{\sqrt{12-3x^2}}\) c) \(y=-x+4\)

- Require the expression under the square root to be nonnegative. - Use the sign before the radical to determine the range. - Apply the chain rule to find the derivative. - Use the point and derivative value to write the tangent line. - Interpret an unbounded derivative near an endpoint geometrically.

1. a) Require \(2x+6\ge0\), so \(D_f=[-3,\infty)\). Since \(\sqrt{2x+6}\ge0\) and grows without bound, the range is \((-\infty,4]\). 2. b) Differentiate: \(f'(x)=-\frac{1}{\sqrt{2x+6}}\). 3. At \(x=\frac{3}{2}\), \(f\left(\frac{3}{2}\right)=1\) and \(f'\left(\frac{3}{2}\right)=-\frac{1}{3}\). 4. Using point-slope form, \(y-1=-\frac{1}{3}\left(x-\frac{3}{2}\right)\), so \(t:y=-\frac{1}{3}x+\frac{3}{2}\). 5. c) As \(x\to-3^+\), \(\sqrt{2x+6}\to0^+\), so \(f'(x)\to-\infty\). The graph has a vertical tangent at its endpoint \((-3,4)\).

a) \(D_f=[-3,\infty)\); range: \((-\infty,4]\) b) \(t:y=-\frac{1}{3}x+\frac{3}{2}\) c) \(\lim_{x\to-3^+}f'(x)=-\infty\); the graph has a vertical tangent at \((-3,4)\).

- Identify which operations occur inside and outside the radical. - Find an x-intercept by setting the function equal to \(0\). - Isolate the radical before squaring. - Differentiate the square-root function to obtain its slope function. - Set the derivative equal to the specified slope.

1. a) Require \(x+1\ge0\), so \(D_h=[-1,\infty)\). Starting with \(w(x)=\sqrt{x}\), shift \(1\) unit left, vertically stretch by \(2\), reflect across the x-axis, and shift \(3\) units up. 2. b) Set \(h(x)=0\): \(3-2\sqrt{x+1}=0\), so \(\sqrt{x+1}=\frac{3}{2}\). Then \(x+1=\frac{9}{4}\), giving \(x=\frac{5}{4}\). The intercept is \(\left(\frac{5}{4},0\right)\). 3. c) Differentiate: \(h'(x)=-\frac{1}{\sqrt{x+1}}\). Set \(h'(x_0)=-\frac{1}{2}\), so \(\sqrt{x_0+1}=2\), and \(x_0=3\).

a) \(D_h=[-1,\infty)\); shift left \(1\), vertically stretch by \(2\), reflect across the x-axis, and shift up \(3\) b) \(\left(\frac{5}{4},0\right)\) c) \(x_0=3\)

- Match each parameter to a horizontal shift, vertical scale, or vertical shift. - Rewrite the square root as a rational power before differentiating if helpful. - Parallel lines have equal slopes. - Substitute the x-coordinate into \(f\) to find the point's y-coordinate.

1. a) Shift the square-root graph \(4\) units left, vertically shrink it by a factor of \(\frac{1}{2}\), and shift it \(3\) units down. 2. b) By the chain rule, \(f'(x)=\frac{1}{4\sqrt{x+4}}\). 3. c) Parallel lines have equal slopes, so set \(f'(x)=\frac{1}{8}\). Then \(\frac{1}{4\sqrt{x+4}}=\frac{1}{8}\), so \(\sqrt{x+4}=2\), giving \(x=0\). 4. Since \(f(0)=\frac{1}{2}\sqrt{4}-3=-2\), the point is \(P=(0,-2)\).

a) Shift left \(4\), vertically shrink by \(\frac{1}{2}\), and shift down \(3\) b) \(f'(x)=\frac{1}{4\sqrt{x+4}}\) c) \(P=(0,-2)\)

- Even powers do not change when \(x\) is replaced by \(-x\). - Use the chain rule for the square-root function. - A smooth graph symmetric about the y-axis has a horizontal tangent where it crosses that axis. - Determine the parity of the derivative of an even function.

1. a) \(g(-x)=\sqrt{(-x)^4+2(-x)^2+5}=\sqrt{x^4+2x^2+5}=g(x)\). Thus, \(g\) is even, so its graph is symmetric about the y-axis. 2. b) By the chain rule, \(g'(x)=\frac{4x^3+4x}{2\sqrt{x^4+2x^2+5}}=\frac{2x^3+2x}{\sqrt{x^4+2x^2+5}}\). 3. The derivative of an even differentiable function is odd. Therefore, \(g'(-x)=-g'(x)\), and at \(x=0\), \(g'(0)=-g'(0)\), which implies \(g'(0)=0\).

a) Since \(g(-x)=g(x)\), the graph is symmetric about the y-axis. b) \(g'(x)=\frac{2x^3+2x}{\sqrt{x^4+2x^2+5}}\), and \(g'(0)=0\) because the derivative of an even differentiable function is odd.

- A square-root radicand must be nonnegative. - Set a function equal to \(0\) to find an x-intercept. - Set the two functions equal to find a shared point. - Use each derivative to find the corresponding tangent slope. - Write each tangent line in point-slope form.

1. a) Require each radicand to be nonnegative. Thus, \(D_f=(-\infty,12]\) and \(D_g=[-4,\infty)\). The x-intercepts are \((12,0)\) and \((-4,0)\), respectively. 2. b) Set the functions equal: \(12-x=x+4\), so \(x=4\). Then \(y=\sqrt{8}=2\sqrt{2}\), giving \(S=(4,2\sqrt{2})\). 3. c) The derivatives are \(f'(x)=-\frac{1}{2\sqrt{12-x}}\) and \(g'(x)=\frac{1}{2\sqrt{x+4}}\). 4. At \(x=4\), the slopes are \(-\frac{\sqrt{2}}{8}\) and \(\frac{\sqrt{2}}{8}\). 5. Using point-slope form, the tangent lines are \(t_f:y=-\frac{\sqrt{2}}{8}x+\frac{5\sqrt{2}}{2}\) and \(t_g:y=\frac{\sqrt{2}}{8}x+\frac{3\sqrt{2}}{2}\).

a) \(D_f=(-\infty,12]\), with x-intercept \((12,0)\); \(D_g=[-4,\infty)\), with x-intercept \((-4,0)\) b) \(S=(4,2\sqrt{2})\) c) \(t_f:y=-\frac{\sqrt{2}}{8}x+\frac{5\sqrt{2}}{2}\); \(t_g:y=\frac{\sqrt{2}}{8}x+\frac{3\sqrt{2}}{2}\)

- Require the square-root radicand to be nonnegative. - Find the graph point at the given x-coordinate. - Apply the chain rule to the radical term. - A line is determined by one point and its slope.

1. a) Require \(7-x\ge0\), so \(D=(-\infty,7]\). 2. At \(x=3\), \(f(3)=2\sqrt{4}+3=7\). 3. By the chain rule, \(f'(x)=1-\frac{1}{\sqrt{7-x}}\), so \(f'(3)=\frac{1}{2}\). 4. Using point-slope form, \(y-7=\frac{1}{2}(x-3)\), so \(y=\frac{1}{2}x+\frac{11}{2}\).

a) \(D=(-\infty,7]\) b) \(y=\frac{1}{2}x+\frac{11}{2}\)

- Require the expression under the radical to be nonnegative. - A tangent slope is a value of the derivative. - Set the derivative equal to the specified slope and solve. - Substitute the resulting x-coordinate into the original function.

1. a) Require \(2x+2\ge0\), so the domain is \([-1,\infty)\). 2. Differentiate: \(g'(x)=\frac{3}{\sqrt{2x+2}}-1\). 3. Set the derivative equal to \(\frac{1}{2}\): \(\frac{3}{\sqrt{2x+2}}-1=\frac{1}{2}\). Then \(\sqrt{2x+2}=2\), so \(x=1\). 4. Evaluate \(g(1)=3\sqrt{4}-1=5\). Thus, \(P=(1,5)\).

a) \([-1,\infty)\) b) \(P=(1,5)\)

- Set \(x=0\) for the y-intercept and \(y=0\) for the x-intercept. - The derivative gives the tangent slope. - Relate the slope to the tangent of the inclination angle. - Read transformations from the operations inside and outside the square root.

1. a) The y-intercept is \((0,g(0))=(0,2)\). For the x-intercept, solve \(4-2\sqrt{x+1}=0\), giving \(x=3\). Thus, the x-intercept is \((3,0)\). 2. b) Differentiate: \(g'(x)=-\frac{1}{\sqrt{x+1}}\). At \(x=0\), the tangent slope is \(-1\). Since \(\tan(135^\circ)=-1\), the inclination angle is \(135^\circ\). 3. c) Starting with \(w(x)=\sqrt{x}\), shift \(1\) unit left, vertically stretch by \(2\), reflect across the x-axis, and shift \(4\) units up.

a) y-intercept \((0,2)\); x-intercept \((3,0)\) b) \(g'(0)=-1=\tan(135^\circ)\), so the inclination angle is \(135^\circ\). c) Shift left \(1\), vertically stretch by \(2\), reflect across the x-axis, and shift up \(4\).

- Require the square-root radicand to be nonnegative. - Replace \(x\) with \(-x\) to test symmetry. - Apply the chain rule to differentiate. - Analyze the signs of the numerator and denominator near each endpoint. - An unbounded derivative magnitude indicates a vertical tangent.

1. a) Require \(9-x^2\ge0\), so \(D_f=[-3,3]\). Since \(f(-x)=f(x)\), the graph is symmetric about the y-axis. It is not symmetric about the x-axis because it contains only nonnegative y-values. 2. b) By the chain rule, \(f'(x)=-\frac{2x}{\sqrt{9-x^2}}\). 3. As \(x\to3^-\), the numerator approaches \(-6\) and the denominator approaches \(0^+\), so \(f'(x)\to-\infty\). 4. As \(x\to-3^+\), the numerator approaches \(6\) and the denominator approaches \(0^+\), so \(f'(x)\to\infty\). 5. Therefore, the graph has vertical tangent lines at \((-3,0)\) and \((3,0)\).

a) \(D_f=[-3,3]\); the graph is symmetric about the y-axis and not symmetric about the x-axis. b) \(f'(x)=-\frac{2x}{\sqrt{9-x^2}}\); \(f'(x)\to-\infty\) as \(x\to3^-\), and \(f'(x)\to\infty\) as \(x\to-3^+\). Both endpoint tangents are vertical.

- Set the radicand greater than or equal to \(0\) to find the domain. - Use the derivative and graph point to write the tangent line. - Examine the radical denominator as \(x\) approaches each endpoint. - A derivative that becomes unbounded does not have a finite endpoint value.

1. a) Require \(100-4x^2\ge0\), so \(x^2\le25\) and the domain is \([-5,5]\). The x-intercepts are \((-5,0)\) and \((5,0)\). 2. Differentiate: \(g'(x)=-\frac{4x}{\sqrt{100-4x^2}}\). 3. b) At \(x=3\), \(g(3)=8\) and \(g'(3)=-\frac{3}{2}\). Thus, \(y-8=-\frac{3}{2}(x-3)\), or \(y=-\frac{3}{2}x+\frac{25}{2}\). 4. c) As \(x\to5^-\), \(g'(x)\to-\infty\). As \(x\to-5^+\), \(g'(x)\to\infty\). 5. The endpoint slopes do not approach finite real values, so the function is not differentiable at the endpoints; the graph has vertical tangents there.

a) Domain \([-5,5]\); x-intercepts \((-5,0)\) and \((5,0)\) b) \(y=-\frac{3}{2}x+\frac{25}{2}\) c) The slope approaches \(-\infty\) at \(5\) from the left and \(\infty\) at \(-5\) from the right, so \(g\) is not differentiable at either endpoint.

- Rewrite the square root as a power before applying the chain rule. - The derivative's denominator determines where the derivative is defined. - For the one-sided limit, track the sign and size of the denominator as the endpoint is approached.

1. Write \(f(x)=(10-2x)^{1/2}-2\). By the chain rule, \(f'(x)=\frac{1}{2}(10-2x)^{-1/2}(-2)=-\frac{1}{\sqrt{10-2x}}\). 2. The derivative is defined when \(10-2x>0\), so the domain of \(f'\) is \(( -\infty,5)\). 3. As \(x\to5^-\), the positive denominator \(\sqrt{10-2x}\) approaches \(0\), so \(f'(x)\to-\infty\). 4. The tangent direction becomes vertical as the graph approaches the endpoint \((5,-2)\), so the graph has a vertical tangent there.

a) \(f'(x)=-\frac{1}{\sqrt{10-2x}}\), with domain \(( -\infty,5)\) b) \(\lim_{x\to5^-}f'(x)=-\infty\) c) The graph has a vertical tangent at the endpoint \((5,-2)\).

- Keep the point of tangency as \((x_0,f(x_0))\) instead of choosing a number for \(x_0\). - Find the derivative before writing the general point-slope equation. - A y-intercept is found by setting the line's input to zero.

1. The chain rule gives \(f'(x)=\frac{4}{\sqrt{8x}}\). 2. At \(x_0>0\), the tangent is \(y=\frac{4}{\sqrt{8x_0}}(x-x_0)+\sqrt{8x_0}\). Since \(\frac{4x_0}{\sqrt{8x_0}}=\frac12\sqrt{8x_0}\), this simplifies to \(y=\frac{4}{\sqrt{8x_0}}x+\frac12\sqrt{8x_0}\). 3. Setting \(x=0\) gives the y-intercept \(\frac12\sqrt{8x_0}=\frac12 f(x_0)\), as required.

a) \(y=\frac{4}{\sqrt{8x_0}}x+\frac12\sqrt{8x_0}\). b) The y-intercept is \(\left(0,\frac12 f(x_0)\right)\) for every \(x_0>0\).

- The normal slope is the negative reciprocal of the tangent slope. - Set the normal line equal to \(0\) to find its x-intercept. - Compare the resulting x-coordinate with \(x_0\). - A point on the x-axis is as far from the origin as the absolute value of its x-coordinate.

1. Differentiate: \(g'(x)=\frac{1}{\sqrt{2x}}\). 2. The normal slope at \(x_0\) is \(m_n=-\sqrt{2x_0}\). 3. The normal equation is \(y=-\sqrt{2x_0}(x-x_0)+\sqrt{2x_0}\). 4. a) Set \(y=0\): \(0=-\sqrt{2x_0}(x_n-x_0)+\sqrt{2x_0}\). Since \(x_0>0\), divide by \(\sqrt{2x_0}\) to obtain \(x_n=x_0+1\). 5. b) For \(x_0=4\), \(x_n=5\), so \(S=(5,0)\). The distance \(OS\) is \(5\) units.

a) \(x_n=x_0+1\) b) \(S=(5,0)\), and \(OS=5\) units

- A y-intercept has x-coordinate \(0\). - The tangent slope is the derivative value at the point. - Relate a line's slope to the tangent of its inclination angle. - The normal slope is the negative reciprocal of the tangent slope.

1. The y-intercept is \(P=(0,h(0))=(0,6)\). 2. By the chain rule, \(h'(x)=-\frac{3}{2\sqrt{4-x}}\), so the tangent slope at \(x=0\) is \(m_t=-\frac{3}{4}\). 3. The inclination angle \(\alpha\) satisfies \(\tan(\alpha)=-\frac{3}{4}\) with \(0^\circ<\alpha<180^\circ\). Thus, \(\alpha\approx143.13^\circ\). 4. The normal slope is \(m_n=-\frac{1}{m_t}=\frac{4}{3}\). Through \((0,6)\), the normal line is \(y=\frac{4}{3}x+6\).

Normal line: \(y=\frac{4}{3}x+6\) Tangent inclination angle: \(\alpha\approx143.13^\circ\)

- The radicand must be nonnegative. - Use the chain rule for the square root. - The derivative value gives the slope. - Isolate and square a radical equation only after checking signs.

1. Require \(2ax-a^2\geq0\). Since \(a>0\), this is equivalent to \(x\geq\frac{a}{2}\). Thus, the domain is \(\left[\frac{a}{2},\infty\right)\). 2. Apply the chain rule: \(f_a'(x)=\frac{a}{\sqrt{2ax-a^2}}\). 3. Set \(f_a'(2)=\frac{1}{2}\): \(\frac{a}{\sqrt{4a-a^2}}=\frac{1}{2}\). 4. Squaring gives \(a^2=\frac{1}{4}(4a-a^2)\), so \(5a^2-4a=0\). 5. Since \(a>0\), \(a=\frac{4}{5}\). The value \(x=2\) lies in the corresponding domain, so the solution is valid.

a) Domain: \(\left[\frac{a}{2},\infty\right)\) b) \(f_a'(x)=\frac{a}{\sqrt{2ax-a^2}}\) c) \(a=\frac{4}{5}\)

- Require the square-root radicand to be nonnegative. - Use the zeros of the quadratic to make a sign chart. - Apply the chain rule to the composite function. - Simplify the factor of \(2\) after differentiating.

1. Require \(x^2-6x+5\ge0\). Since \(x^2-6x+5=(x-1)(x-5)\), the expression is nonnegative for \(x\le1\) or \(x\ge5\). 2. Therefore, \(D_f=(-\infty,1]\cup[5,\infty)\). 3. By the chain rule, \(f'(x)=\frac{2x-6}{2\sqrt{x^2-6x+5}}=\frac{x-3}{\sqrt{x^2-6x+5}}\), for \(x<1\) or \(x>5\).

\(D_f=(-\infty,1]\cup[5,\infty)\); \(f'(x)=\frac{x-3}{\sqrt{x^2-6x+5}}\) on the interior of \(D_f\)

- Read horizontal changes from inside the radical and vertical changes from outside it. - Apply the chain rule to differentiate. - Substitute the given input into the derivative to find the slope. - Use the endpoint value and long-run behavior to determine the range.

1. Shift the square-root graph \(4\) units left, vertically stretch it by \(2\), reflect it across the x-axis, and shift it \(3\) units up. 2. By the chain rule, \(f'(x)=-\frac{1}{\sqrt{x+4}}\), for \(x>-4\). Thus, \(f'(0)=-\frac{1}{2}\). 3. At the left endpoint, \(f(-4)=3\), which is the greatest value. As \(x\to\infty\), \(f(x)\to-\infty\). Therefore, the range is \((-\infty,3]\).

1. Shift left \(4\), vertically stretch by \(2\), reflect across the x-axis, and shift up \(3\). 2. \(f'(x)=-\frac{1}{\sqrt{x+4}}\); \(f'(0)=-\frac{1}{2}\) 3. \((-\infty,3]\)

- Parallel lines have equal slopes. - Set the derivative equal to the target slope. - Apply the chain rule to the square-root function. - Check solutions after squaring because signs may have been lost. - Use point-slope form once the point of tangency is known.

1. Differentiate: \(f'(x)=-\frac{x}{\sqrt{100-x^2}}\). 2. A parallel tangent must have slope \(\frac{3}{4}\), so solve \(-\frac{x}{\sqrt{100-x^2}}=\frac{3}{4}\). 3. Squaring gives \(16x^2=9(100-x^2)\), so \(x^2=36\). Checking the original equation shows that the positive slope requires \(x=-6\). 4. The point of tangency is \((-6,f(-6))=(-6,8)\). 5. Using point-slope form, \(y-8=\frac{3}{4}(x+6)\), so \(y=\frac{3}{4}x+\frac{25}{2}\).

\(y=\frac{3}{4}x+\frac{25}{2}\)

- Perpendicular nonvertical lines have slopes whose product is \(-1\). - Set the derivative equal to the required tangent slope. - Check the sign after squaring. - Use the point and slope to write the line equation.

1. Differentiate: \(g'(x)=\frac{x}{\sqrt{x^2+12}}\). 2. A line perpendicular to a line of slope \(-2\) has slope \(\frac{1}{2}\). 3. Solve \(\frac{x}{\sqrt{x^2+12}}=\frac{1}{2}\). Squaring gives \(4x^2=x^2+12\), so \(x^2=4\). Checking the original equation gives \(x=2\). 4. The point of tangency is \((2,g(2))=(2,4)\). 5. Using point-slope form, \(y-4=\frac{1}{2}(x-2)\), so \(y=\frac{1}{2}x+3\).

\(y=\frac{1}{2}x+3\)

- A square-root radicand must be nonnegative. - Apply the chain rule: outer derivative times inner derivative. - The tangent slope at a point is the derivative value there.

1. a) Require \(25-x^2\ge0\), so \(D=[-5,5]\). 2. b) Apply the chain rule: \(f'(x)=2\cdot\frac{1}{2\sqrt{25-x^2}}(-2x)=-\frac{2x}{\sqrt{25-x^2}}\), for \(-5<x<5\). 3. c) \(f'(3)=-\frac{6}{\sqrt{16}}=-\frac{3}{2}\).

a) \(D=[-5,5]\) b) \(f'(x)=-\frac{2x}{\sqrt{25-x^2}}\) for \(-5<x<5\) c) \(-\frac{3}{2}\)

- Apply the chain rule and multiply by the derivative of the radicand. - A tangent line requires the graph point and derivative value. - Use point-slope form to write the line.

1. a) By the chain rule, \(g'(x)=\frac{4x}{2\sqrt{2x^2+17}}=\frac{2x}{\sqrt{2x^2+17}}\). 2. b) At \(x=2\), \(g(2)=5\) and \(g'(2)=\frac{4}{5}\). 3. Using point-slope form, \(y-5=\frac{4}{5}(x-2)\), so \(y=\frac{4}{5}x+\frac{17}{5}\).

a) \(g'(x)=\frac{2x}{\sqrt{2x^2+17}}\) b) \(y=\frac{4}{5}x+\frac{17}{5}\)

- Require the logarithm’s input to be positive. - Differentiate the outer logarithm and multiply by the inner derivative. - Substitute \(x=2\) into the derivative.

1. The logarithm requires \(12-4x>0\). Solving gives \(x<3\), so \(D_g=(-\infty, 3)\). 2. Apply the chain rule: \(g'(x)=\frac{-4}{12-4x}=\frac{1}{x-3}\). 3. Evaluate the derivative at \(x=2\): \(g'(2)=\frac{1}{2-3}=-1\).

a) \(D_g=(-\infty, 3)\) b) \(g'(x)=\frac{1}{x-3}\) c) The slope is \(-1\).

- Which real inputs make the argument of the logarithm positive? - Identify the inside expression before applying the chain rule. - Check whether a logarithm identity works for negative values of \(x\) as well as positive ones.

1. The logarithm requires \(5x^4>0\). Since \(x^4>0\) exactly when \(x\ne0\), the domain is \(\mathbb{R}\setminus\{0\}\). 2. Apply the chain rule to the logarithm: \(f'(x)=\frac{1}{5x^4}\cdot20x^3=\frac{4}{x}\), for \(x\ne0\). 3. The logarithm properties give \(f(x)=\ln5+4\ln|x|\) for every \(x\ne0\). If \(x>0\), then \((\ln|x|)'=1/x\). If \(x<0\), write \(\ln|x|=\ln(-x)\); the chain rule gives \((\ln|x|)'=\frac{1}{-x}(-1)=1/x\). Both branches give \(f'(x)=4/x\), agreeing with part 2.

1. \(\mathbb{R}\setminus\{0\}\) 2. \(f'(x)=\frac{20x^3}{5x^4}=\frac{4}{x}\), for \(x\ne0\) 3. \(f(x)=\ln5+4\ln|x|\). For \(x>0\), \((\ln|x|)'=1/x\); for \(x<0\), \(\ln|x|=\ln(-x)\), whose derivative is \(\frac{1}{-x}(-1)=1/x\). Thus both branches give \(f'(x)=4/x\).

- Determine when the logarithm's argument is positive. - Apply the chain rule as outer derivative times inner derivative. - Compare the domains of \(\ln(x^4)\) and \(\ln x\). - Consider how absolute value produces a valid identity for negative inputs.

1. The logarithm requires \(x^4>0\), which holds exactly when \(x\ne0\). Therefore, \(D_f=\mathbb{R}\setminus\{0\}\). 2. Apply the chain rule: \(f'(x)=\frac{1}{x^4}(4x^3)=\frac{4}{x}\), for \(x\ne0\). 3. The derivative formula is correct on the entire domain, but the stated identity is not. The expression \(4\ln x\) is defined only for \(x>0\), while \(\ln(x^4)\) is also defined for \(x<0\). A domain-valid identity is \(\ln(x^4)=4\ln|x|\) for \(x\ne0\).

a) \(D_f=\mathbb{R}\setminus\{0\}\) b) \(f'(x)=\frac{4}{x}\), for \(x\ne0\) c) The derivative result is correct, but the identity \(\ln(x^4)=4\ln x\) is valid only for \(x>0\). On the full domain, use \(\ln(x^4)=4\ln|x|\).

- Find where the squared expression equals zero. - Apply the chain rule carefully. - A tangent line requires a point and a slope. - Evaluate both the function and derivative at \(x=1\).

1. The logarithm’s argument is zero only when \(2x-4=0\), so \(x=2\). It is positive for every other real input. Therefore, \(D_k=\mathbb{R}\setminus\{2\}\). 2. Differentiate using the chain rule: \(k'(x)=\frac{4(2x-4)}{(2x-4)^2}=\frac{2}{x-2}\). At \(x=1\), \(k(1)=\ln4\) and \(k'(1)=-2\). Thus, the tangent line is \(y=-2(x-1)+\ln4=-2x+2+\ln4\).

a) \(D_k=\mathbb{R}\setminus\{2\}\) b) \(y=-2x+2+\ln4\)

- Distinguish the order of operations in a composition from multiplication or division. - Differentiate the three basic functions before applying combined rules. - The first composed function also has an external cosine factor.

1. The first function is a product of a composition and \(v\): \(f_1(x)=(u\circ w)(x)v(x)\). 2. The second combines composition and addition: \(f_2(x)=(v\circ w)(x)+u(x)\). 3. The third is a quotient: \(f_3(x)=\frac{w(x)}{u(x)}\). 4. The basic derivatives are \(u'(x)=e^x\), \(v'(x)=-\sin x\), and \(w'(x)=2x\). 5. Use the chain and product rules to get \(f_1'(x)=e^{x^2+3}(2x\cos x-\sin x)\). 6. Use the chain and sum rules to get \(f_2'(x)=e^x-2x\sin(x^2+3)\).

a) \(f_1(x)=(u\circ w)(x)v(x)\) b) \(f_2(x)=(v\circ w)(x)+u(x)\) c) \(f_3(x)=\frac{w(x)}{u(x)}\) d) \(u'(x)=e^x\), \(v'(x)=-\sin x\), \(w'(x)=2x\) e) \(f_1'(x)=e^{x^2+3}(2x\cos x-\sin x)\); \(f_2'(x)=e^x-2x\sin(x^2+3)\)

- Identify the inside expression of each outer function. - First find the basic derivatives before composing them. - Both a logarithm and a square root can restrict a derivative's domain.

1. The compositions and quotient are \(g_1=a\circ c\), \(g_2=b\circ a\), and \(g_3(x)=\frac{b(x)}{c(x)}\). 2. Differentiate the basic functions: \(a'(x)=\frac{1}{x}\) for \(x>0\), \(b'(x)=\frac{1}{2\sqrt{x}}\) for \(x>0\), and \(c'(x)=-2x\). 3. The chain rule gives \(g_1'(x)=\frac{-2x}{5-x^2}=\frac{2x}{x^2-5}\). Since the logarithm needs \(5-x^2>0\), this derivative exists for \(-\sqrt{5}<x<\sqrt{5}\). 4. The chain rule gives \(g_2'(x)=\frac{1}{2\sqrt{\ln x}}\cdot\frac{1}{x}=\frac{1}{2x\sqrt{\ln x}}\). This derivative exists for \(x>1\).

a) \(g_1=a\circ c\) b) \(g_2=b\circ a\) c) \(g_3(x)=\frac{b(x)}{c(x)}\) d) \(a'(x)=\frac{1}{x}\), \(b'(x)=\frac{1}{2\sqrt{x}}\), \(c'(x)=-2x\) e) \(g_1'(x)=\frac{2x}{x^2-5}\) for \(-\sqrt{5}<x<\sqrt{5}\); \(g_2'(x)=\frac{1}{2x\sqrt{\ln x}}\) for \(x>1\).

- The input of the outer logarithm must be positive. - Determine when \(\ln x>0\). - Treat \(\ln x\) as the inner function. - Apply the chain rule step by step.

1. The outer logarithm requires \(\ln x>0\). This occurs when \(x>1\), which also satisfies the inner logarithm’s requirement \(x>0\). Thus, \(D_g=(1, \infty)\). 2. Apply the chain rule: \(g'(x)=\frac{1}{\ln x}\cdot\frac{1}{x}=\frac{1}{x\ln x}\).

\(D_g=(1, \infty)\) \(g'(x)=\frac{1}{x\ln x}\)

- Determine when the fraction inside the logarithm is positive. - Remember that \(x=0\) is excluded. - Use the quotient property of logarithms. - When rewriting a logarithm of an even power, will your expression still work for negative inputs?

1. The logarithm's argument must be positive. Since \(x^2>0\) for \(x\ne0\), the numerator must satisfy \(x+2>0\). Therefore, the domain is \((-2,0)\cup(0,\infty)\). 2. Rewrite using logarithm properties: \(f(x)=\ln(x+2)-2\ln|x|\) on this domain. 3. Apply the chain rule to the shifted logarithm: \(f'(x)=\frac{1}{x+2}-\frac{2}{x}=-\frac{x+4}{x(x+2)}\).

Domain: \((-2,0)\cup(0,\infty)\) \(f'(x)=-\frac{x+4}{x(x+2)}\)

- Use the fact that \(\ln1=0\). - Apply the chain rule to find the derivative. - A tangent line requires a point and a slope. - Use point-slope form.

1. To find the zero, solve \(\ln(4-x)=0\). Thus, \(4-x=1\), so \(x=3\). 2. Differentiate: \(f'(x)=-\frac{1}{4-x}=\frac{1}{x-4}\). At \(x=3\), \(f(3)=0\) and \(f'(3)=-1\). Therefore, the tangent line is \(y=-(x-3)=-x+3\).

Zero: \(x=3\) Tangent line: \(y=-x+3\)

- Recall the domain restriction for a logarithm. - Use the chain rule to differentiate \(g\). - At a local extremum of \(u\), what is the value of \(u'\)? - Determine which logarithm input produces an output of \(0\).

1. Statement (1) is false. If \(u(x_0)=0\), then \(\ln(u(x_0))=\ln(0)\) is undefined, so \(x_0\) is not even in the domain of \(g\). 2. By the chain rule, \(g'(x)=\frac{u'(x)}{u(x)}\). At a local extremum \(x_0\) of \(u\), \(u'(x_0)=0\). Because \(x_0\) is in the domain of \(g\), \(u(x_0)>0\), so \(g'(x_0)=0\). Statement (2) is true. 3. If \(g(x_0)=0\), then \(\ln(u(x_0))=0\). Exponentiating gives \(u(x_0)=e^0=1\). Statement (3) is true.

(1) False (2) True (3) True

- Factor the quadratic inside the logarithm. - Determine where that expression is positive. - Apply the chain rule. - Cancel common factors only after preserving the domain restriction.

1. Factor the logarithm’s argument: \(x^2+4x+4=(x+2)^2\). This expression is positive for every real \(x\) except \(x=-2\). Thus, \(D_f=\mathbb{R}\setminus\{-2\}\). 2. Apply the chain rule: \(f'(x)=\frac{2x+4}{x^2+4x+4}\). Factor and simplify: \(f'(x)=\frac{2(x+2)}{(x+2)^2}=\frac{2}{x+2}\), for \(x\ne-2\).

\(D_f=\mathbb{R}\setminus\{-2\}\) \(f'(x)=\frac{2}{x+2}\)

- Check the logarithm's argument in each function. - For the square-root expression, the quantity under the radical cannot be negative. - Count how many nested functions appear in part b before differentiating. - Compare the original domain with the derivative's denominator restrictions.

1. For a), \(x^2+2x+2=(x+1)^2+1>0\), so \(h\) is defined for all real \(x\). 2. The chain rule gives \(h'(x)=\frac{2x+2}{x^2+2x+2}\), again for all real \(x\). 3. For b), the square root requires \(\ln(4x+1)\geq0\), which means \(4x+1\geq1\), or \(x\geq0\). 4. Differentiate both layers: \(k'(x)=\frac{1}{2\sqrt{\ln(4x+1)}}\cdot\frac{4}{4x+1}=\frac{2}{(4x+1)\sqrt{\ln(4x+1)}}\). The derivative exists for \(x>0\), not at the endpoint \(0\).

a) Domain: all real \(x\). \(h'(x)=\frac{2x+2}{x^2+2x+2}\), for all real \(x\). b) Domain: \([0,\infty)\). \(k'(x)=\frac{2}{(4x+1)\sqrt{\ln(4x+1)}}\), for \(x>0\).

- Does the order of two function compositions matter? - For each exponential, identify its complete exponent before differentiating. - Evaluate both derivatives at the same input only after finding their formulas.

1. The compositions are \(h(x)=e^{e^{-x}}\) and \(j(x)=e^{-e^x}\). 2. The chain rule gives \(h'(x)=e^{e^{-x}}(-e^{-x})=-e^{-x}e^{e^{-x}}\). 3. Similarly, \(j'(x)=e^{-e^x}(-e^x)=-e^xe^{-e^x}\). 4. At \(x=0\), \(h'(0)=-e\) and \(j'(0)=-e^{-1}\). They are not equal.

a) \(h(x)=e^{e^{-x}}\), \(j(x)=e^{-e^x}\). b) \(h'(x)=-e^{-x}e^{e^{-x}}\), \(j'(x)=-e^xe^{-e^x}\). c) \(h'(0)=-e\), \(j'(0)=-\frac{1}{e}\); not equal.

- Differentiate the exponent as well as the outer exponential. - To find an input from a slope, first isolate the exponential term. - What values can an exponential function take, and what would a horizontal tangent require?

1. The exponent \(x/2\) has derivative \(1/2\), so \(h'(x)=-\frac{1}{2}e^{x/2}\). 2. Solving \(-\frac{1}{2}e^{x/2}=-3\) gives \(e^{x/2}=6\), hence \(x=2\ln(6)\). 3. The exponential is positive for every real input, so \(h'(x)<0\) everywhere and cannot equal zero. No horizontal tangent exists.

a) \(h'(x)=-\frac{1}{2}e^{x/2}\). b) \(x=2\ln(6)\). c) No; \(h'(x)<0\) for every real \(x\).

- Substitute the coordinates of the given point into the function. - Rewrite the square root as a power and apply the chain rule. - The derivative gives the tangent slope. - Use point-slope form for the tangent line.

1. Substitute \(A(2, 4)\): \(4=\sqrt{k-4}\). Squaring gives \(16=k-4\), so \(k=20\). 2. For \(f_{20}(x)=\sqrt{20-x^2}\), the chain rule gives \(f_{20}'(x)=-\frac{x}{\sqrt{20-x^2}}\). 3. The slope at \(x=2\) is \(f_{20}'(2)=-\frac{2}{4}=-\frac{1}{2}\). 4. Using point-slope form through \((2, 4)\), \(y-4=-\frac{1}{2}(x-2)\). Therefore, \(y=-\frac{1}{2}x+5\).

a) \(k=20\) b) \(y=-\frac{1}{2}x+5\)

- For trigonometric functions, keep the original input when differentiating. - Track both the negative coefficient and inner derivative in part b. - For the radical, check its domain and whether the derivative's denominator is nonzero.

1. For a), differentiate the cosine and its input: \(f'(x)=-5\sin(5x+2)\), for all real \(x\). 2. For b), use the constant factor and the inner derivative: \(g'(x)=-2(0.4)\cos(0.4x)=-0.8\cos(0.4x)\), for all real \(x\). 3. For c), the chain rule gives \(\frac{d}{dx}\sqrt{2x+1}=\frac{2}{2\sqrt{2x+1}}=\frac{1}{\sqrt{2x+1}}\). Therefore \(h'(x)=\frac{1}{\sqrt{2x+1}}+\cos x\), for \(x>-\frac{1}{2}\).

a) \(f'(x)=-5\sin(5x+2)\), for all real \(x\). b) \(g'(x)=-0.8\cos(0.4x)\), for all real \(x\). c) \(h'(x)=\frac{1}{\sqrt{2x+1}}+\cos x\), for \(x>-\frac{1}{2}\).

- Identify the inner and outer functions. - Apply the chain rule to differentiate the composition. - Use the exact sine value for the special angle in part 2.

1. The outer function is cosine, and the inner function is \(x^2\). By the chain rule, \(f'(x)=-\sin(x^2)(2x)=-2x\sin(x^2)\). 2. Substitute \(x=\sqrt{\frac{\pi}{3}}\): \(f'\left(\sqrt{\frac{\pi}{3}}\right)=-2\sqrt{\frac{\pi}{3}}\sin\left(\frac{\pi}{3}\right)\). Since \(\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\), \(f'\left(\sqrt{\frac{\pi}{3}}\right)=-2 \cdot \sqrt{\frac{\pi}{3}} \cdot \left(\frac{\sqrt{3}}{2}\right)=-\sqrt{\pi}\).

1) \(f'(x)=-2x\sin(x^2)\) 2) The exact slope is \(-\sqrt{\pi}\).

- A horizontal tangent has slope zero. - Identify which operation acts on \(x\) first and which acts on the result. - Use the unit circle or known zeros of sine and cosine.

1. One decomposition is \(v(x)=\cos(x)\) and \(u(t)=t^2\). 2. By the chain rule, \(g'(x)=2\cos(x)(-\sin(x))=-2\sin(x)\cos(x)\). Equivalently, \(g'(x)=-\sin(2x)\). 3. A horizontal tangent occurs where \(g'(x)=0\). Thus, \(-2\sin(x)\cos(x)=0\), so either \(\sin(x)=0\) or \(\cos(x)=0\). On \([0, \pi]\), these equations give \(x=0\), \(x=\frac{\pi}{2}\), and \(x=\pi\).

1) \(v(x)=\cos(x)\) and \(u(t)=t^2\) 2) \(g'(x)=-2\sin(x)\cos(x)=-\sin(2x)\) 3) \(x=0\), \(x=\frac{\pi}{2}\), and \(x=\pi\)

- Evaluate the composition from the inside out. - Apply the chain rule once for each layer of the composition. - The derivative is the product of the derivative factors from the outermost function to the innermost function. - A logarithm property can provide a useful check.

1. Work from the inside out: \(v(w(x))=\sqrt{x^2+5}\), so \(f(x)=\ln\left(\sqrt{x^2+5}\right)\). Equivalently, \(f(x)=\frac{1}{2}\ln(x^2+5)\). 2. Applying the chain rule to all three layers gives \(f'(x)=\frac{1}{\sqrt{x^2+5}}\left(\frac{1}{2\sqrt{x^2+5}}\right)(2x)\) \(=\frac{x}{x^2+5}\).

1) \(f(x)=\ln\left(\sqrt{x^2+5}\right)=\frac{1}{2}\ln(x^2+5)\) 2) \(f'(x)=\frac{x}{x^2+5}\)

- List the operations in the order in which they are performed on an input. - Recall that \(\cos^4(z)\) means \((\cos(z))^4\). - Remember the negative sign in the derivative of cosine. - Combine the constant factors after applying the chain rule.

1. One decomposition is \(w(x)=3x-1\), \(v(t)=\cos(t)\), and \(u(t)=t^4\). 2. Apply the chain rule through all three layers: \(h'(x)=4\cos^3(3x-1)[-\sin(3x-1)]\cdot 3\) \(=-12\cos^3(3x-1)\sin(3x-1)\).

1) \(w(x)=3x-1\), \(v(t)=\cos(t)\), and \(u(t)=t^4\) 2) \(h'(x)=-12\cos^3(3x-1)\sin(3x-1)\)

- Build the composition from the innermost function outward. - Substitute the entire output of one function into the next function. - Apply the chain rule once for each nested layer. - Recall the derivatives of cosine and the exponential function.

1. First, \(q(r(x))=\cos^2(x)-4\). Therefore, \(f(x)=e^{\cos^2(x)-4}\). 2. Differentiate the exponent using the chain rule: \(\frac{d}{dx}[\cos^2(x)-4]=2\cos(x)(-\sin(x))=-2\sin(x)\cos(x)\). 3. Apply the chain rule to the exponential function: \(f'(x)=-2\sin(x)\cos(x)e^{\cos^2(x)-4}\). Since \(2\sin(x)\cos(x)=\sin(2x)\), this can also be written as \(f'(x)=-\sin(2x)e^{\cos^2(x)-4}\).

a) \(f(x)=e^{\cos^2(x)-4}\) b) \(f'(x)=-2\sin(x)\cos(x)e^{\cos^2(x)-4}=-\sin(2x)e^{\cos^2(x)-4}\)

- Describe the function as three operations performed in sequence. - Recall the derivatives of the square-root and exponential functions. - Simplify after multiplying the three derivative factors. - Rewrite the expression using exponent rules to check your result.

1. One decomposition is \(u(t)=\sqrt{t}\), \(v(t)=e^t\), and \(w(x)=x^2+4\). 2. Their derivatives are \(u'(t)=\frac{1}{2\sqrt{t}}\), \(v'(t)=e^t\), and \(w'(x)=2x\). 3. By the chain rule, \(g'(x)=\frac{1}{2\sqrt{e^{x^2+4}}}e^{x^2+4}(2x)\) \(=x\sqrt{e^{x^2+4}}\). 4. As a check, \(g(x)=e^{\frac{1}{2}x^2+2}\), whose derivative is \(g'(x)=xe^{\frac{1}{2}x^2+2}\), the same result.

\(g'(x)=x\sqrt{e^{x^2+4}}=xe^{\frac{1}{2}x^2+2}\)

- In the quotient rule, the derivative of the constant numerator is zero. - Rewrite \(\frac{1}{z}\) as \(z^{-1}\) for the chain-rule method. - Track the negative sign and the new exponent carefully.

1. Using the quotient rule with numerator \(5\) and denominator \(x^2+3\), \(f'(x)=\frac{0(x^2+3)-5(2x)}{(x^2+3)^2}\) \(=-\frac{10x}{(x^2+3)^2}\). 2. Rewrite the function as \(f(x)=5(x^2+3)^{-1}\). Using the chain rule, \(f'(x)=5[-(x^2+3)^{-2}](2x)\) \(=-\frac{10x}{(x^2+3)^2}\). The two methods agree.

\(f'(x)=-\frac{10x}{(x^2+3)^2}\)

- How does a polynomial's leading term change when its input is another polynomial? - Find the degrees of \(u'(v(x))\) and \(v'(x)\) separately before combining them. - Compare the degree of a composition with the degree of a product.

1. The composition of polynomials of degrees \(n\) and \(m\) has degree \(nm\). 2. The chain rule gives \(f'(x)=u'(v(x))v'(x)\). The factors have degrees \((n-1)m\) and \(m-1\) (a nonzero constant counts as degree \(0\)). Their sum is \(nm-1\). 3. The product \(h(x)=u(x)v(x)\) has degree \(n+m\). Equality requires \(nm=n+m\), or \((n-1)(m-1)=1\). For positive integers, the only solution is \(n=m=2\).

1) \(\deg(f)=nm\). 2) \(f'(x)=u'(v(x))v'(x)\); its factor degrees sum to \(nm-1\). 3) \(n=2\), \(m=2\).

- Start by writing the chain rule for a composition of two unspecified functions. - How do the degrees of the two derivative factors relate to the original degrees? - Check that reversing an ordered pair gives a separate possibility when the degrees differ.

1. The chain rule gives \(f'(x)=p'(q(x))q'(x)\). Its two factor degrees are \((n-1)m\) and \(m-1\), so \(\deg(f')=nm-1\). 2. Given degree \(11\), we need \(nm=12\). With both factors at least \(2\), the ordered pairs are \((2,6)\), \((3,4)\), \((4,3)\), and \((6,2)\).

a) \(f'(x)=p'(q(x))q'(x)\), with factor degrees \((n-1)m\) and \(m-1\); hence \(\deg(f')=nm-1\). b) \((2,6)\), \((3,4)\), \((4,3)\), \((6,2)\).

- Use the chain rule for \(f\). - A tangent line needs a point and a derivative value. - Equal exponential expressions with the same base have equal exponents. - Compare the two derivative values at the intersection input.

1. By the chain rule, \(f'(x)=2e^{2x-4}\). At \(x=2\), \(f(2)=1\) and \(f'(2)=2\). Thus, the tangent line is \(y-1=2(x-2)\), or \(y=2x-3\). 2. At an intersection, \(e^{2x-4}=e^x\). Since the bases are equal and the exponential function is one-to-one, \(2x-4=x\), so \(x=4\). 3. Since \(g'(x)=e^x\), \(g'(4)=e^4\). Also, \(f'(4)=2e^{2\cdot4-4}=2e^4\). Therefore, \(f'(4)=2g'(4)\).

a) \(f'(x)=2e^{2x-4}\); tangent line: \(y=2x-3\) b) The x-coordinate of \(S\) is \(4\). c) \(f'(4)=2e^4=2g'(4)\).

- First find the derivative using the chain rule. - The slope at a point is the value of the first derivative there. - Use radians for the trigonometric inputs. - Treat the power in part c as the outer function.

1. For a), \(f'(x)=\frac{x}{\sqrt{x^2+9}}\), so \(f'(4)=\frac{4}{\sqrt{25}}=\frac{4}{5}=0.8\). 2. For b), \(f'(x)=2\cos(2x)\), so \(f'\left(\frac{\pi}{6}\right)=2\cos\left(\frac{\pi}{3}\right)=1\). 3. For c), \(f'(x)=3(\cos(x)+1)^2(-\sin(x))\), so \(f'\left(\frac{\pi}{2}\right)=3\cdot 1^2\cdot(-1)=-3\).

a) \(f'(4)=0.8\) b) \(f'\left(\frac{\pi}{6}\right)=1\) c) \(f'\left(\frac{\pi}{2}\right)=-3\)

- Identify the inner and outer functions. - Expand the cube by multiplying or using the binomial theorem. - Expand or factor one derivative expression to compare the two forms.

1. Using the chain rule, \(f'(x)=3(0.5x^2-2)^2(x)=3x(0.5x^2-2)^2\). 2. Expanding first gives \(f(x)=0.125x^6-1.5x^4+6x^2-8\). Therefore, \(f'(x)=0.75x^5-6x^3+12x\). 3. Expand the chain-rule result: \(3x(0.5x^2-2)^2=3x(0.25x^4-2x^2+4)\) \(=0.75x^5-6x^3+12x\). Thus, the results are equivalent.

1) \(f'(x)=3x(0.5x^2-2)^2\) 2) \(f'(x)=0.75x^5-6x^3+12x\) 3) Expanding the first expression gives the second expression.

- Rewrite radicals as fractional powers when useful. - A fraction with a constant numerator can be rewritten using a negative exponent. - Differentiate the terms of a sum separately.

1. For a), \(g'(x)=\frac{1}{2\sqrt{x^2+2x+5}}(2x+2)\) \(=\frac{x+1}{\sqrt{x^2+2x+5}}\). This derivative is defined for all real \(x\). 2. For b), rewrite \(h(x)=10(3x-1)^{-4}\). Then \(h'(x)=10[-4(3x-1)^{-5}]\cdot 3\) \(=-\frac{120}{(3x-1)^5}\), which is defined for \(x\ne\frac{1}{3}\). 3. For c), differentiate each term: \(k'(x)=\frac{1}{2\sqrt{2x}}\cdot 2+2(x^3-1)(3x^2)\) \(=\frac{1}{\sqrt{2x}}+6x^2(x^3-1)\). This derivative is defined for \(x>0\).

a) \(g'(x)=\frac{x+1}{\sqrt{x^2+2x+5}}\), for all real \(x\) b) \(h'(x)=-\frac{120}{(3x-1)^5}\), for \(x\ne\frac{1}{3}\) c) \(k'(x)=\frac{1}{\sqrt{2x}}+6x^2(x^3-1)\), for \(x>0\)

- Treat the square root as the outer function. - Use the quotient rule on the expression under the radical. - Multiply the outer derivative by the inner derivative. - Track signs carefully when expanding the quotient-rule numerator.

1. Let \(r(x)=\frac{3x+2}{x^2+1}\). The outer function is \(\sqrt{r}\), whose derivative is \(\frac{1}{2\sqrt{r}}\). 2. By the quotient rule, \(r'(x)=\frac{3(x^2+1)-(3x+2)(2x)}{(x^2+1)^2}\) \(=\frac{-3x^2-4x+3}{(x^2+1)^2}\). 3. Apply the chain rule: \(f'(x)=\frac{1}{2\sqrt{\frac{3x+2}{x^2+1}}}\left(\frac{-3x^2-4x+3}{(x^2+1)^2}\right)\) \(=\frac{-3x^2-4x+3}{2(x^2+1)^2\sqrt{\frac{3x+2}{x^2+1}}}\). The derivative is defined for \(x>-\frac{2}{3}\).

\(f'(x)=\frac{-3x^2-4x+3}{2(x^2+1)^2\sqrt{\frac{3x+2}{x^2+1}}}\), for \(x>-\frac{2}{3}\)

- Find the point and slope needed for a tangent-line equation. - Use point-slope form at \(x_0=4\). - Apply the chain rule when differentiating the exponential function. - Set \(y=0\) for the x-intercept and \(x=0\) for the y-intercept. - Use the intercept distances as the legs of the right triangle.

1. Evaluate the function: \(f(4)=4e^{-1}=\frac{4}{e}\). Differentiate using the chain rule: \(f'(x)=-e^{-0.25x}\), so \(f'(4)=-\frac{1}{e}\). 2. Use point-slope form: \(y-\frac{4}{e}=-\frac{1}{e}(x-4)\). Therefore, \(t: y=-\frac{1}{e}x+\frac{8}{e}\). 3. For the y-intercept, set \(x=0\), giving \(\left(0, \frac{8}{e}\right)\). For the x-intercept, set \(y=0\), giving \(x=8\), so the intercept is \((8, 0)\). 4. The triangle has legs \(8\) and \(\frac{8}{e}\). Its area is \(A=\frac{1}{2}\cdot8\cdot\frac{8}{e}=\frac{32}{e}\approx11.77\) square units.

1. \(t: y=-\frac{1}{e}x+\frac{8}{e}\) 2. x-intercept: \((8, 0)\); y-intercept: \(\left(0, \frac{8}{e}\right)\) 3. \(A=\frac{32}{e}\approx11.77\) square units

- Work with \(x_0\) and \(k\) symbolically in part 1. - At an x-intercept, the y-coordinate is \(0\). - Factor out \(e^{kx_0}\) from the intercept equation. - Use the result from part 1 for parts 2 and 3. - The origin has x-coordinate \(0\).

1. Differentiate: \(f_k'(x)=ke^{kx}\). The tangent line at \(x_0\) is \(y=ke^{kx_0}(x-x_0)+e^{kx_0}\). 2. At the x-intercept, \(y=0\). Thus \(0=e^{kx_0}[k(x_s-x_0)+1]\). Since \(e^{kx_0}>0\), \(k(x_s-x_0)+1=0\), so \(x_s=x_0-\frac{1}{k}\). 3. Substitute \(x_0=3\) and \(x_s=1\): \(1=3-\frac{1}{k}\). Therefore, \(\frac{1}{k}=2\), so \(k=0.5\). 4. A line through the origin has x-intercept \(x_s=0\). With \(k=0.5\), \(0=x_0-\frac{1}{0.5}=x_0-2\), so \(x_0=2\).

1. \(x_s=x_0-\frac{1}{k}\) 2. \(k=0.5\) 3. \(x_0=2\)

- Parallel lines have the same slope. - Use the chain rule to differentiate the logarithmic function. - Setting the derivative equal to the line's slope produces a quadratic equation. - Use point-slope form for each tangent line.

1. Rewrite the given line as \(y=\frac{4}{5}x+2\), so its slope is \(\frac{4}{5}\). 2. Differentiate using the chain rule: \(f'(x)=\frac{2x}{x^2+1}\). 3. Set the derivative equal to the required slope: \(\frac{2x}{x^2+1}=\frac{4}{5}\). This gives \(4x^2-10x+4=0\), or \(2x^2-5x+2=0\). 4. Factor: \((2x-1)(x-2)=0\). Thus \(x=\frac{1}{2}\) or \(x=2\). 5. The corresponding points are \(P_1\left(\frac{1}{2}, \ln\left(\frac{5}{4}\right)\right)\) and \(P_2(2, \ln 5)\). 6. Using slope \(\frac{4}{5}\), the tangent lines are \(y-\ln\left(\frac{5}{4}\right)=\frac{4}{5}\left(x-\frac{1}{2}\right)\) and \(y-\ln 5=\frac{4}{5}(x-2)\).

Points: \(P_1\left(\frac{1}{2}, \ln\left(\frac{5}{4}\right)\right)\) and \(P_2(2, \ln 5)\) Tangent lines: \(y=\frac{4}{5}x-\frac{2}{5}+\ln\left(\frac{5}{4}\right)\) and \(y=\frac{4}{5}x-\frac{8}{5}+\ln 5\)

- Matching a tangent means matching both the height and the slope at that input. - Write a general cubic with four unknown coefficients. Which two can be found immediately? - Use the remaining point and the horizontal tangent to determine the other coefficients.

1. The chain rule gives \(f'(x)=-2e^{-x/2}\), so \(f'(0)=-2\), while \(f(0)=4\). 2. Write \(p(x)=ax^3+bx^2+cx+d\) and \(p'(x)=3ax^2+2bx+c\). Tangency at zero gives \(d=4\) and \(c=-2\). 3. The horizontal tangent gives \(p'(-1)=3a-2b-2=0\), so \(3a-2b=2\). The point \((1,2)\) gives \(a+b-2+4=2\), so \(a+b=0\). 4. Solve these two equations: \(a=\frac25\), \(b=-\frac25\). Substituting all coefficients gives \(p(x)=\frac25x^3-\frac25x^2-2x+4\).

a) \(f'(0)=-2\). b) \(p(x)=\frac25x^3-\frac25x^2-2x+4\).

- A line through the origin has equation \(y=mx\). - At the point of tangency, the line's slope equals the derivative. - Use the chain rule to differentiate \(\ln(x^2)\). - Determine how many solutions the equation for \(u\) has.

1. Differentiate using the chain rule: \(f'(x)=\frac{2}{x}\). 2. Let the point of tangency have x-coordinate \(u\neq0\). A tangent line through the origin has slope \(\frac{2}{u}\), so its equation is \(y=\frac{2}{u}x\). 3. The point \((u, \ln(u^2))\) must lie on this line. Therefore, \(\ln(u^2)=\frac{2}{u}u=2\). 4. Solving \(\ln(u^2)=2\) gives \(u^2=e^2\), so \(u=e\) or \(u=-e\). These are the only two solutions, so exactly two such tangent lines exist. 5. Their slopes are \(\frac{2}{e}\) and \(-\frac{2}{e}\). Thus the tangent lines are \(y=\frac{2}{e}x\) and \(y=-\frac{2}{e}x\).

a) The two points of tangency occur at \(u=e\) and \(u=-e\). b) \(y=\frac{2}{e}x\) and \(y=-\frac{2}{e}x\)

- Evaluate a composition from the inside out. - Use the chain rule for a function of the form \(f(g(x))\). - A horizontal tangent occurs where the derivative is zero. - Use the zero-product property on the two derivative factors.

1. From the graph, \(f(2)=3\) and \(f(4)=4\). Therefore, \(h(1)=f(1^2+3)=f(4)=4\). 2. By the chain rule, \(h'(x)=2x f'(x^2+3)\). 3. A horizontal tangent occurs where \(2x f'(x^2+3)=0\). One possibility is \(2x=0\), which gives \(x=0\). The graph of \(f\) has a horizontal tangent only when its input is \(4\), so the other possibility is \(x^2+3=4\). Thus, \(x^2=1\), giving \(x=-1\) or \(x=1\). Therefore, the graph of \(h\) has horizontal tangents at \(x=-1\), \(x=0\), and \(x=1\).

a) \(f(2)=3\), \(f(4)=4\), and \(h(1)=4\) b) \(h'(x)=2x f'(x^2+3)\); the horizontal tangents occur at \(x=-1\), \(x=0\), and \(x=1\).

- Use the horizontal asymptote to identify \(c\). - Use the y-intercept to find \(a\). - Substitute the graph's zero to find \(b\). - Differentiate using the chain rule and evaluate at \(0\) and \(1\).

1. The graph approaches the horizontal asymptote \(y=4\) as \(x\to-\infty\), so \(c=4\). The y-intercept is \((0, 2)\), so \(a+4=2\) and \(a=-2\). 2. The graph has a zero at \(x=1\). Thus, \(-2e^b+4=0\), so \(e^b=2\) and \(b=\ln2\). 3. Therefore, \(f(x)=-2e^{(\ln2)x}+4\), and \(f'(x)=-2\ln2\,e^{(\ln2)x}\). Hence, \(f'(0)=-2\ln2\) and \(f'(1)=-4\ln2=2f'(0)\).

a) \(c=4\), \(a=-2\) b) \(b=\ln2\) c) \(f'(0)=-2\ln2\) and \(f'(1)=-4\ln2=2f'(0)\)

- Find the point by substituting the specified input. - Rewrite the denominator as a negative power before using the chain rule. - For the acute angle, use the size of the slope, not its sign.

1. Evaluate the function: \(f(2)=\frac{8}{8}=1\), so the tangent point is \((2,1)\). 2. The chain rule gives \(f'(x)=-\frac{16x}{(x^2+4)^2}\). Thus \(f'(2)=-\frac{1}{2}\). 3. The tangent line is \(y-1=-\frac{1}{2}(x-2)\), or \(y=-\frac{x}{2}+2\). 4. Its acute angle with the x-axis satisfies \(\tan\beta=\frac{1}{2}\), so \(\beta\approx26.6^\circ\).

a) \(y=-\frac{x}{2}+2\) b) \(\beta\approx26.6^\circ\)

- A logarithm requires a positive argument. - Apply the chain rule to \(\ln(u(x))\). - Use the monotonicity and zero of \(f\) to determine where \(f(x)>0\). - Express \(g'\) in terms of \(f\) and \(f'\), then simplify the equation. - Read the required function value from the graph.

1. For part a, require \(3x-5>0\), so \(D_h=\left(\frac{5}{3}, \infty\right)\). The zero satisfies \(3x-5=1\), giving \(x=2\). By the chain rule, \(h'(x)=\frac{3}{3x-5}\). 2. Since \(f\) is strictly increasing and has its only zero at \(-2\), \(f(x)>0\) exactly when \(x>-2\). Therefore, \(D_g=(-2, \infty)\). Also, \(g'(x)=\frac{f'(x)}{f(x)}\). The equation \(\frac{f'(x)}{f(x)}=\frac{1}{2}f'(x)\) can be divided by \(f'(x)>0\), giving \(f(x)=2\). From the graph, \(f(x)=2\) at \(x=2\).

a) \(D_h=\left(\frac{5}{3}, \infty\right)\), zero at \(x=2\), and \(h'(x)=\frac{3}{3x-5}\) b) 1. \(D_g=(-2, \infty)\) 2. \(x=2\)

- The derivative of a sine function with a variable input includes the inner derivative. - Compare the distance of each graph's highest point from its midline. - Maximum slope means the largest derivative value.

1. By the chain rule, \(h'(x)=2\cos(2x)\). 2. The original function has amplitude \(1\), while its derivative has amplitude \(2\). Thus p represents \(h\), and q represents \(h'\). 3. The greatest slope of \(h\) is the maximum of \(h'(x)\). Since \(2\cos(2x)\leq2\), this slope is \(2\). 4. On \([0,\pi]\), \(\cos(2x)=1\) at \(x=0\) and \(x=\pi\).

a) \(h'(x)=2\cos(2x)\) b) Graph p is \(h\), and graph q is \(h'\). c) The maximum slope is \(2\), at \(x=0\) and \(x=\pi\).

- Evaluate the inner function first, then use its output as the input of the outer function. - Apply the chain rule \(f^{\prime}(x)=u^{\prime}(v(x))v^{\prime}(x)\). - Read each required slope from the line or tangent shown. - Use point-slope form to write the tangent equation.

1. From the graph, \(v(1)=2\), and then \(u(2)=-1\). Therefore, \(f(1)=u(v(1))=-1\). 2. By the chain rule, \(f^{\prime}(1)=u^{\prime}(v(1))v^{\prime}(1)=u^{\prime}(2)v^{\prime}(1)\). 3. The graph of \(v\) is a line with slope \(1\), so \(v^{\prime}(1)=1\). The displayed tangent to \(u\) at \(x=2\) has slope \(1\), so \(u^{\prime}(2)=1\). Thus, \(f^{\prime}(1)=1\). 4. The tangent through \((1, -1)\) with slope \(1\) is \(y+1=x-1\), or \(y=x-2\). 5. Its intercepts are \((2, 0)\) and \((0, -2)\). The triangle has area \(\frac{1}{2}\cdot2\cdot2=2\) square units.

a) \(f(1)=-1\) and \(f^{\prime}(1)=1\) b) \(2\) square units

- A horizontal tangent occurs where the derivative is zero. - Apply the chain rule to the square of a function. - Look for both zeros of \(f\) and points where the graph of \(f\) has a horizontal tangent.

1. By the chain rule, \(g'(x)=2f(x)f'(x)\). 2. A horizontal tangent occurs where \(g'(x)=0\). By the zero-product property, this happens when \(f(x)=0\) or \(f'(x)=0\). 3. The graph shows that \(f(x)=0\) at \(x=1\) and \(x=5\). 4. The graph of \(f\) has a horizontal tangent at its vertex, \(x=3\), so \(f'(3)=0\). Therefore, the graph of \(g\) has horizontal tangents at \(x=1\), \(x=3\), and \(x=5\).

The horizontal tangents occur at \(x=1\), \(x=3\), and \(x=5\).

- Evaluate the inner function before the outer function. - Find another x-value where the inner function has the same output. - Use the chain rule to find the tangent slope. - Check the tangent’s intercepts before calculating the triangle’s area.

1. From the graph, \(v(1)=1\), so \(f(1)=u(v(1))=u(1)=1\). 2. The graph of \(v\) also has \(v(3)=1\), so \(f(3)=u(1)=1\). 3. By the chain rule, \(f^{\prime}(1)=u^{\prime}(v(1))v^{\prime}(1)=u^{\prime}(1)v^{\prime}(1)\). 4. The tangent to \(u\) has slope \(\frac{1}{2}\), and the tangent to \(v\) at \(x=1\) has slope \(-1\). Thus, \(f^{\prime}(1)=\frac{1}{2}\cdot(-1)=-0.5\). 5. The tangent through \((1, 1)\) is \(y-1=-0.5(x-1)\), or \(y=-0.5x+1.5\). 6. Its intercepts are \((3, 0)\) and \((0, 1.5)\), so the triangle’s area is \(\frac{1}{2}\cdot3\cdot1.5=2.25\) square units.

a) \(f(1)=1\); another x-value is \(x=3\). b) The tangent is \(y=-0.5x+1.5\), and the area is \(2.25\) square units.

- Evaluate the inner graph first. - Find all inputs of \(u\) that produce the same output. - Use the chain rule for the tangent slope. - Find the tangent line intercepts before computing the triangle area.

1. The graph gives \(v(1)=2\) and \(u(2)=0\), so \(f(1)=0\). 2. The zeros of \(u\) occur at inputs \(0\) and \(2\). Besides \(v(1)=2\), the graph gives \(v(3)=0\). Therefore, the second value is \(x=3\). 3. By the chain rule, \(f'(x)=u'(v(x))v'(x)\). From the graphs, \(v'(x)=-1\) and \(u'(2)=2\), so \(f'(1)=-2\). 4. The tangent line through \((1, 0)\) is \(y=-2(x-1)=-2x+2\). Its intercepts are \((1, 0)\) and \((0, 2)\). The triangle area is \(\frac{1}{2}\cdot1\cdot2=1\).

a) \(f(1)=0\); the second value is \(x=3\). b) \(1\) square unit

- Find the slope at a general x-coordinate a. - Use point-slope form, then set y=0 to locate the x-intercept. - Horizontal distance depends on the x-coordinates, not the heights.

1. The chain rule gives \(f'(x)=0.5e^{0.5x}\). 2. The tangent equation is \(y=0.5e^{0.5a}(x-a)+e^{0.5a}\). 3. At \(Q\), set \(y=0\). Since \(e^{0.5a}>0\), divide by it to get \(0=0.5(x_Q-a)+1\), so \(x_Q=a-2\). 4. The horizontal distance is \(|a-x_Q|=|a-(a-2)|=2\), independent of \(a\).

a) \(y=0.5e^{0.5a}(x-a)+e^{0.5a}\) b) \(x_Q=a-2\) c) The horizontal distance is \(2\) units for every real \(a\).

- Evaluate the function at \(x=0\). - A change in an outside multiplier produces a vertical scaling. - Use the point and slope at \(x=0\) to write the tangent line.

1. Since \(g_k(0)=k\), the y-intercepts give p: \(k=2\) and q: \(k=4\). 2. Doubling \(k\) doubles every function value, which is a vertical stretch by a factor of \(2\). 3. The derivative is \(g_k'(x)=-ke^{-x}\). At \(x=0\), the point is \((0, k)\) and the slope is \(-k\). 4. The tangent line is \(y=-kx+k\). Setting \(y=0\) gives \(x=1\), independent of \(k\).

a) p: \(k=2\); q: \(k=4\) b) Vertical stretch by a factor of \(2\) c) The tangent is \(y=-kx+k\), with x-intercept \(1\).

- The horizontal asymptote identifies the vertical shift. - Use the y-intercept to find \(a\). - Substitute the given point and compare exponents. - Differentiate to find the tangent slope.

1. The horizontal asymptote is \(y=3\), so \(c=3\). The y-intercept is \((0, 1)\), so \(a+3=1\) and \(a=-2\). 2. Substitute \(P(2, 3-2e)\) into \(f(x)=-2e^{bx}+3\): \(3-2e=-2e^{2b}+3\). Thus, \(e=e^{2b}\), so \(b=\frac{1}{2}\). 3. The function is \(f(x)=-2e^{x/2}+3\), so \(f'(x)=-e^{x/2}\). At \(x=0\), the slope is \(-1\), and the point is \((0, 1)\). Therefore, the tangent line is \(y=-x+1\).

a) \(c=3\), \(a=-2\) b) \(b=\frac{1}{2}\) c) \(y=-x+1\)

- The y-intercept occurs when x is zero. - Compare f(-x) and f(x) to test symmetry. - For the tangent, combine the chain-rule slope with the point on the curve. - Set the tangent equation equal to zero for its x-intercept.

1. At \(x=0\), \(f(0)=4\), so the y-intercept is \((0,4)\). 2. For every real \(x\), \(f(-x)=4e^{-(-x)^2/2}=4e^{-x^2/2}=f(x)\), so the graph is symmetric about the y-axis. 3. The chain rule gives \(f'(x)=-4xe^{-x^2/2}\). At \(x=1\), the point is \((1,4e^{-1/2})\) and the slope is \(-4e^{-1/2}\). 4. The tangent line is \(y-4e^{-1/2}=-4e^{-1/2}(x-1)\), or \(y=-4e^{-1/2}x+8e^{-1/2}\). 5. Setting \(y=0\) yields \(x=2\).

a) \((0,4)\) b) \(f(-x)=f(x)\), so there is y-axis symmetry. c) \(y=-4e^{-1/2}x+8e^{-1/2}\) d) \((2,0)\)

- What values can \(f(x)\) take according to the graph? - How does the increasing function \(e^u\) transform inequalities between exponent values? - In the chain rule, which factor determines the sign of \(g'(x)\)?

1. The quadratic function \(f\) is defined for every real number, and \(e^u\) is defined for every real input \(u\). Therefore, the domain of \(g\) is \(\mathbb{R}\). 2. From the graph, \(f\) has a maximum value of \(1\) at \(x=0\), and \(f(x)\to-\infty\) as \(|x|\to\infty\). Because the exponential function is strictly increasing, \(g(x)\leq e\), and \(g(x)>0\). Thus, the range is \((0, e]\). 3. The graph of \(f\) increases for \(x<0\) and decreases for \(x>0\). Since \(e^u\) preserves order, \(g\) has the same monotonicity: it increases on \((-\infty, 0]\) and decreases on \([0, \infty)\). 4. Equivalently, \(g'(x)=f'(x)e^{f(x)}\), and the exponential factor is always positive, so the sign of \(g'\) is the sign of \(f'\).

a) \(\mathbb{R}\) b) \((0, e]\) c) Increasing on \((-\infty, 0]\) and decreasing on \([0, \infty)\)

- Find the output of \(g\) at \(x=0\). Which input does \(f'\) then need? - Read the two slopes from the graphs before combining them.

1. Read \(g(0)=1\) and \(g'(0)=2\) from the line \(g\). 2. In the first graph, the tangent to \(f\) at input \(1\) has slope \(2\), so \(f'(1)=2\). 3. The chain rule gives \(h'(0)=f'(g(0))g'(0)=2\cdot2=4\).

\(h'(0)=4\)

- Which part of the formula determines whether the depth is rising or falling? - Keep the exponential unchanged while finding the derivative of its exponent. - What does a negative change in depth mean for the tank?

1. Differentiate the exponential using the chain rule: \(H'(t)=3e^{-t^2/16}\left(-\frac{t}{8}\right)=-\frac{3t}{8}e^{-t^2/16}\). 2. At \(t=4\), \(H'(4)=-\frac{3}{2}e^{-1}=-\frac{3}{2e}\) inches per hour. 3. The negative sign means the water depth is decreasing at that time.

\(H'(4)=-\frac{3}{2e}\,\text{in./hr}\). The water depth is falling.

- Tangent graphs have equal values and equal slopes at the same input. - Differentiate both functions. - Use the double-angle identity for sine. - Solve the derivative equation first, then use the value equation.

1. At a point of tangency, the function values and derivatives must be equal: \(f(x)=g_c(x)\) and \(f'(x)=g_c'(x)\). 2. The derivatives are \(f'(x)=2\cos x\) and \(g_c'(x)=-2\sin(2x)\). Set them equal and use \(\sin(2x)=2\sin x\cos x\): \(2\cos x=-4\sin x\cos x\), so \(2\cos x(1+2\sin x)=0\). 3. If \(\cos x=0\), then \(\sin x=1\) or \(-1\), and \(\cos(2x)=-1\). For \(\sin x=1\), the value equation gives \(2=-1+c\), so \(c=3\). For \(\sin x=-1\), it gives \(-2=-1+c\), so \(c=-1\). 4. If \(\sin x=-\frac{1}{2}\), then \(\cos(2x)=\frac{1}{2}\). The value equation gives \(-1=\frac{1}{2}+c\), so \(c=-\frac{3}{2}\).

\(c\in\left\{-\frac{3}{2},-1,3\right\}\)

- Tangency requires equal function values and equal derivatives. - Use the chain rule to differentiate \(\sin^2x\). - Factor the derivative equation. - Use \(\sin^2x+\cos^2x=1\) in the remaining case.

1. At a point of tangency, \(f(x)=h_k(x)\) and \(f'(x)=h_k'(x)\). 2. Differentiate: \(f'(x)=2\sin x\cos x\) and \(h_k'(x)=\sin x\). Set the slopes equal: \(\sin x(2\cos x-1)=0\). 3. If \(\sin x=0\), then \(\cos x=1\) or \(-1\). When \(\cos x=1\), the value equation gives \(0=k-1\), so \(k=1\). When \(\cos x=-1\), it gives \(0=k+1\), so \(k=-1\). 4. If \(\cos x=\frac{1}{2}\), then \(\sin^2x=1-\frac{1}{4}=\frac{3}{4}\). The value equation gives \(\frac{3}{4}=k-\frac{1}{2}\), so \(k=\frac{5}{4}\).

\(k\in\left\{-1,1,\frac{5}{4}\right\}\)

- Identify the inner and outer functions in each composition. - Use the chain rule: derivative of the outer function times derivative of the inner function. - Recall how the parameters in \(a\sin(bx+c)+d\) affect amplitude, period, and shifts. - Use the minimum and maximum values of sine to determine the range.

1. The compositions are \(h_1(x)=\sin(mx+c)\) and \(h_2(x)=m\sin(x)+c\). 2. By the chain rule, \(h_1'(x)=m\cos(mx+c)\) and \(h_2'(x)=m\cos(x)\). 3. The amplitude of \(h_1\) remains \(1\). Its period is \(\frac{2\pi}{m}\). Since \(mx+c=m\left(x+\frac{c}{m}\right)\), the graph is shifted horizontally by \(-\frac{c}{m}\): left when \(c>0\) and right when \(c<0\). 4. Since \(-1\le \sin(x)\le 1\) and \(m>0\), the range of \(h_2\) is \([c-m, c+m]\). Set the endpoints equal to the required range: \(c-m=-2\), \(c+m=4\). Adding gives \(2c=2\), so \(c=1\). Then \(m=3\).

1. \(h_1(x)=\sin(mx+c)\); \(h_2(x)=m\sin(x)+c\) 2. \(h_1'(x)=m\cos(mx+c)\); \(h_2'(x)=m\cos(x)\) 3. The period is \(\frac{2\pi}{m}\), the amplitude is \(1\), and the horizontal shift is \(-\frac{c}{m}\). 4. \(m=3\) and \(c=1\)

- Use the identity \(\sin(-z)=-\sin(z)\). - Remember the inner derivative when differentiating \(f(-x)\). - Recall the symmetry identity for cosine. - Consider how the product of two even functions behaves.

1. Since \(f\) is odd, \(f(-x)=-f(x)\). Therefore, \(h(-x)=\sin(f(-x))=\sin(-f(x))=-\sin(f(x))=-h(x)\). Thus, \(h\) is odd. 2. By the chain rule, \(h'(x)=\cos(f(x))f'(x)\). 3. Differentiate \(f(-x)=-f(x)\) with respect to \(x\): \(-f'(-x)=-f'(x)\). Hence, \(f'(-x)=f'(x)\), so \(f'\) is even. 4. Now, \(h'(-x)=\cos(f(-x))f'(-x)\) \(=\cos(-f(x))f'(x)\) \(=\cos(f(x))f'(x)=h'(x)\). Therefore, \(h'\) is even.

a) \(h(-x)=\sin(f(-x))=\sin(-f(x))=-h(x)\), so \(h\) is odd. b) \(h'(x)=\cos(f(x))f'(x)\) c) Differentiating \(f(-x)=-f(x)\) gives \(-f'(-x)=-f'(x)\), so \(f'(-x)=f'(x)\). Thus, \(f'\) is even, and \(h'(-x)=\cos(-f(x))f'(x)=h'(x)\), so \(h'\) is even.

- First treat \(g(h(x))\) as one inner function. - Apply the chain rule again when differentiating that inner function. - Evaluate the table from the inside out: begin with \(h(0)\), then evaluate \(g\), and finally evaluate the needed derivative of \(f\). - Distinguish carefully between function values and derivative values.

1. Applying the chain rule to \(k(x)=f(V(x))\) gives \(k'(x)=f'(V(x))V'(x)\). 2. Since \(V(x)=g(h(x))\), another application of the chain rule gives \(V'(x)=g'(h(x))h'(x)\). Therefore, \(k'(x)=f'(g(h(x)))g'(h(x))h'(x)\). 3. At \(x=0\), the table gives \(h(0)=1\) and \(h'(0)=5\). 4. Then \(g(h(0))=g(1)=0\), and the table gives \(g'(1)=3\) and \(f'(0)=1\). 5. Thus, \(k'(0)=f'(0)g'(1)h'(0)=1\cdot 3\cdot 5=15\).

a) \(k'(x)=f'(g(h(x)))g'(h(x))h'(x)\) b) \(k'(0)=15\)

- Substitute the given point into the function to form one equation. - Use the derivative and the given slope to form a second equation. - Solve the two equations for \(a\) and \(c\). - The expression \(x-c\) controls the horizontal shift, and \(a\) controls vertical scaling.

1. The point condition gives \(a\sqrt{5-c}=4\). 2. Since \(h'(x)=\frac{a}{2\sqrt{x-c}}\), the slope condition gives \(\frac{a}{2\sqrt{5-c}}=1\), or \(a=2\sqrt{5-c}\). 3. Substitute into the point condition: \(2(5-c)=4\). Thus, \(c=3\), and then \(a=2\sqrt{2}\). 4. The graph is shifted \(3\) units right and vertically stretched by a factor of \(2\sqrt{2}\).

a) \(a=2\sqrt{2}\); \(c=3\) b) Shift right \(3\) units and vertically stretch by a factor of \(2\sqrt{2}\).

- A normal slope is the negative reciprocal of the tangent slope when both are finite and nonzero. - Write a general normal equation at \(x_0\), and test whether \(M\) satisfies it. - Treat horizontal, vertical, and endpoint tangents as separate cases. - A tangent parallel to \(y=x\) has slope \(1\). - Check solutions in the original radical equation after squaring.

1. a) Require \(x(12-x)\ge0\), so \(D_g=[0,12]\). For \(0<x<12\), \(g'(x)=\frac{6-x}{\sqrt{12x-x^2}}\). 2. At \(x=2\), \(g(2)=2\sqrt{5}\), and the tangent slope is \(\frac{2}{\sqrt{5}}\). The normal slope is \(-\frac{\sqrt{5}}{2}\), so \(n:y=-\frac{\sqrt{5}}{2}x+3\sqrt{5}\). 3. b) For \(0<x_0<12\) and \(x_0\ne6\), the normal slope is \(m_n=\frac{\sqrt{12x_0-x_0^2}}{x_0-6}\). Substituting \(M=(6,0)\) into the normal equation gives \(0=m_n(6-x_0)+g(x_0)\), which is true. 4. At \(x_0=6\), the tangent is horizontal, so the normal is \(x=6\), which contains \(M\). At the endpoints \((0,0)\) and \((12,0)\), the tangents are vertical, so the normal is the x-axis, which also contains \(M\). 5. c) Parallelism to \(y=x\) requires \(g'(x)=1\). Thus, \(6-x=\sqrt{12x-x^2}\). Squaring gives \(x^2-12x+18=0\), with candidates \(x=6\pm3\sqrt{2}\). 6. The original unsquared equation requires \(x<6\), so \(x=6-3\sqrt{2}\). Then \(g(x)=3\sqrt{2}\), giving \(P=(6-3\sqrt{2},3\sqrt{2})\).

a) \(D_g=[0,12]\); \(n:y=-\frac{\sqrt{5}}{2}x+3\sqrt{5}\) b) Every normal contains \(M=(6,0)\); the interior, center, and endpoint cases all satisfy the claim. c) \(P=(6-3\sqrt{2},3\sqrt{2})\)

- Rewrite a variable base and exponent using the exponential function. - Differentiate the exponent using the product and chain rules. - Apply the chain rule to the outer exponential function.

1. Use \(a^b=e^{b\ln a}\): \(f(x)=e^{x\ln(\ln x)}\). Thus, \(g(x)=x\ln(\ln x)\). 2. Differentiate \(g\) using the product and chain rules: \(g'(x)=\ln(\ln x)+\frac{1}{\ln x}\). 3. Differentiate \(f(x)=e^{g(x)}\): \(f'(x)=e^{g(x)}g'(x)\). Therefore, \(f'(x)=(\ln x)^x\left(\ln(\ln x)+\frac{1}{\ln x}\right)\).

a) \(f(x)=e^{x\ln(\ln x)}\) b) \(f'(x)=(\ln x)^x\left(\ln(\ln x)+\frac{1}{\ln x}\right)\)

- Turn the given points and slope into equations for the three constants. - The input of the cosine includes a factor of two; account for it when differentiating. - Try using the two function values to eliminate one coefficient before using the slope.

1. The chain rule gives \(f'(x)=a\cos(x)-2b\sin(2x)\). 2. From \(f(0)=4\), obtain \(b+c=4\). From \(f(\frac{\pi}{2})=1\), obtain \(a-b+c=1\); therefore \(a=2b-3\). 3. The slope condition is \(\frac{a\sqrt{2}}{2}-2b=\sqrt{2}-5\). Substitute \(a=2b-3\) to obtain \(b(\sqrt{2}-2)=\frac52(\sqrt{2}-2)\), so \(b=\frac52\). 4. Then \(a=2\) and \(c=\frac32\). Each original condition is satisfied.

a) \(f'(x)=a\cos(x)-2b\sin(2x)\). b) \(a=2\), \(b=\frac52\), \(c=\frac32\).

- At each candidate input, use the function value and derivative to write a tangent line. - The chain rule helps differentiate the sine of a multiple of \(x\). - Compare each tangent's height at the sensor's horizontal position. - Keep exact values until the last step so rounding does not change the comparison.

1. Use the chain rule: \(g'(x)=0.5\cos(0.5x)\). The tangent released at \(x=a\) has height \(g(a)+g'(a)(10-a)\) when it reaches \(x=10\). 2. At \(x_1=\frac{3\pi}{2}\), the predicted height is \(2+\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{4}\left(10-\frac{3\pi}{2}\right)\approx0.838\). 3. At \(x_2=2\pi\), the predicted height is \(2-\frac{1}{2}(10-2\pi)=\pi-3\approx0.142\). 4. At \(x_3=\frac{5\pi}{2}\), the predicted height is \(2-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{4}\left(10-\frac{5\pi}{2}\right)\approx0.534\). 5. Only the tangent at \(x_2=2\pi\) matches the sensor height to three decimal places.

At \(x_1=\frac{3\pi}{2}\), the predicted height is \(0.838\); at \(x_2=2\pi\), it is \(0.142\); at \(x_3=\frac{5\pi}{2}\), it is \(0.534\). The closest match is \(x_2=2\pi\).

- Apply the chain rule to the composition. - A product is zero when at least one factor is zero. - Identify horizontal tangents on both graphs. - Find where \(v\) takes the input values at which \(u\) has horizontal tangents, using its equation for exact values.

1. The graph gives \(v(1)=1\) and \(u(1)=-1\), so \(g(1)=-1\). 2. By the chain rule, \(g'(x)=u'(v(x))v'(x)\). Since \(u\) has a horizontal tangent at input \(1\), \(u'(1)=0\). Therefore, \(g'(1)=0\). 3. The product \(u'(v(x))v'(x)\) is zero when either factor is zero. The graph of \(v\) has a horizontal tangent at \(x=0\), giving one zero. 4. The graph of \(u\) has horizontal tangents at inputs \(-1\) and \(1\). Solve \(v(x)=1\) and \(v(x)=-1\) using \(v(x)=2-x^2\). These give \(x=\pm1\) and \(x=\pm\sqrt{3}\), respectively. 5. Thus the five zeros are \(-\sqrt{3}\), \(-1\), \(0\), \(1\), and \(\sqrt{3}\).

a) \(g(1)=-1\) and \(g'(1)=0\) b) \(x\in\{-\sqrt{3}, -1, 0, 1, \sqrt{3}\}\)