Chapter 4

Differentiate the functions given in Problems with respect to the independent variable. $$ f(t)=t^{3} e^{-2}+t+e^{-1} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\frac{1}{(1-x)^{2}} \text { at } a=0 $$

Differentiate the functions with respect to the independent variable. \(g(t)=\sqrt{t^{2}+\sqrt{t+1}}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=2 \cos \left(x^{3}-3 x\right) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=\frac{x}{e^{x}+e^{-x}} $$

Use the product rule to find the derivative with respect to the independent variable. $$ h(s)=\left(4-3 s^{2}+4 s^{3}\right)^{2} $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\frac{1}{2} x^{2} e^{3}-x^{4} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\ln (1+2 x) \text { at } a=0 $$

Compute \(f(c+h)-f(c)\) at the indicated point. $$ f(x)=-2 x+1 ; c=2 $$

Differentiate the functions with respect to the independent variable. \(g(t)=\left(\frac{t}{t-3}\right)^{3}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\sin ^{3}\left(x^{2}-3\right) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=e^{\sin (3 x)} $$

In Problems \(17-20\), apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\left(3 x^{2}-2\right)(x-1), \text { at } x=1 $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(s)=s^{3} e^{3}+3 e $$

Use (4.12) to find the derivative of the inverse at the indicated point. Let \(f(x)=\ln (\sin x), 0

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\ln (1+2 x) \text { at } a=0 $$

Compute \(f(c+h)-f(c)\) at the indicated point. $$ f(x)=3 x^{2} ; c=1 $$

Differentiate the functions with respect to the independent variable. \(h(s)=\left(\frac{2 s^{2}}{s+1}\right)^{4}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\cos ^{2}\left(x^{2}-1\right) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=e^{\cos (4 x)} $$

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=(1-2 x)(1+2 x), \text { at } x=2 $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\frac{x}{e}+e^{2} x+e $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\log x \text { at } a=1 $$

Compute \(f(c+h)-f(c)\) at the indicated point. $$ f(x)=\sqrt{x} ; c=4 $$

Differentiate the functions with respect to the independent variable. \(f(r)=\left(r^{2}-r\right)^{3}\left(r+3 r^{3}\right)^{-4}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=3 \sin ^{2} x^{2} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=e^{\sin \left(x^{2}-1\right)} $$

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=4\left(2 x^{4}+3 x\right)\left(4-2 x^{2}\right), \text { at } x=-1 $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=20 x^{3}-4 x^{6}+9 x^{8} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\log \left(1+x^{2}\right) \text { at } a=0 $$

Compute \(f(c+h)-f(c)\) at the indicated point. $$ f(x)=\frac{1}{x} ; c=-2 $$

Differentiate the functions with respect to the independent variable. \(h(s)=\frac{2(3-s)^{2}}{s^{2}+(7 s-1)^{2}}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-\sin ^{2}\left(2 x^{3}-1\right) $$

Apply the product rule to find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\left(3 x^{3}-3\right)\left(2-2 x^{2}\right), \text { at } x=0 $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\frac{x^{3}}{15}-\frac{x^{4}}{20}+\frac{2}{15} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=e^{x} \text { at } a=0 $$

Use the formal definition of the derivative to find the derivative of \(y=5 x^{2}\) at \(x=-1\). (b) Show that the point \((-1,5)\) is on the graph of \(y=5 x^{2}\), and find the equation of the tangent line at the point \((-1,5)\). (c) Graph \(y=5 x^{2}\) and the tangent line at the point \((-1,5)\) in the same coordinate system.

Differentiate the functions with respect to the independent variable. \(h(x)=\sqrt[5]{3-x^{4}}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=4 \cos x^{2}-2 \cos ^{2} x $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=\sin \left(e^{x}\right) $$

In Problems \(21-24\), apply the product rule to find the normal line, in slope- intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=(1-x)\left(2-x^{2}\right), \text { at } x=2 $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\pi x^{3}-\frac{1}{\pi}+\frac{x}{\pi} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=e^{2 x} \text { at } a=0 $$

Use the formal definition to find the derivative of \(y=\) \(-2 x^{2}\) at \(x=1\) (b) Show that the point \((1,-2)\) is on the graph of \(y=-2 x^{2}\), and find the equation of the tangent line at the point \((1,-2)\). (c) Graph \(y=-2 x^{2}\) and the tangent line at the point \((1,-2)\) in the same coordinate system.

Differentiate the functions with respect to the independent variable. \(h(x)=\sqrt[3]{1-2 x}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-5 \cos \left(2-x^{3}\right)+2 \cos ^{3}(x-4) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=\cos \left(e^{x}\right) $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=\pi x e^{2}-\frac{x^{2} \pi}{e} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=e^{-x} \text { at } a=0 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt[7]{x^{2}-2 x+1}\)