Chapter 4

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

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=5(1-2 x)(x+1)-3, \text { at } x=0 $$

Differentiate $$ f(x)=a x^{3} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=e^{-3 x} \text { at } a=0 $$

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

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt[4]{2-4 x^{2}}\)

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

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=\cos \left(3 x-e^{x^{2}-1}\right) $$

Differentiate $$ f(x)=x^{3}+a $$

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

Use the formal definition to find the derivative of $$ y=\sqrt{x} $$

Differentiate the functions with respect to the independent variable. \(g(s)=\left(3 s^{7}-7 s\right)^{3 / 2}\)

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

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=\exp [x-\sin x] $$

In Problems \(25-28\), apply the product rule repeatedly to find the derivative of \(y=f(x) .\) $$ f(x)=(2 x-1)(3 x+4)(1-x) $$

Differentiate $$ f(x)=a x^{2}-2 a $$

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

Use the formal definition to find the derivative of $$ f(x)=\frac{1}{x+1} $$

Differentiate the functions with respect to the independent variable. \(h(t)=\left(t^{4}-5 t\right)^{5 / 2}\)

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

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

Apply the product rule repeatedly to find the derivative of \(y=f(x) .\) $$ f(x)=(x-3)(2-3 x)(5-x) $$

Differentiate $$ f(x)=a^{2} x^{4}-2 a x^{2} $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=(1+x)^{-n}\) at \(a=0\). (Assume that \(n\) is a positive integer.)

Find the equation of the tangent line to the curve \(y=3 x^{2}\) at the point \((1,3)\).

Differentiate the functions with respect to the independent variable. \(h(t)=\left(3 t+\frac{3}{t}\right)^{2 / 5}\)

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

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(s)=\exp \left[\sec s^{2}\right] $$

Apply the product rule repeatedly to find the derivative of \(y=f(x) .\) $$ f(x)=(x-3)\left(2 x^{2}+1\right)\left(1-x^{2}\right) $$

Differentiate $$ h(s)=r s^{2}-r $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=(1-x)^{-n}\) at \(a=0\). (Assume that \(n\) is a positive integer.)

Find the equation of the tangent line to the curve \(y=2 / x\) at the point \((2,1)\).

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

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

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(s)=\exp \left[\tan s^{3}\right] $$

Apply the product rule repeatedly to find the derivative of \(y=f(x) .\) $$ f(x)=(2 x+1)\left(4-x^{2}\right)\left(1+x^{2}\right) $$

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

Find the equation of the tangent line to the curve \(y=\sqrt{x}\) at the point \((4,2)\).

Differentiate $$ f(x)=(a x+1)^{3} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

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

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

Differentiate $$ f(x)=a(x-1)(2 x-1) $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \left(2 x^{3}-x\right) $$

Differentiate $$ f(x)=r s^{2} x^{3}-r x+s $$

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\left(1+\frac{1}{x}\right)^{1 / 4} \text { at } a=1 $$

Find the equation of the tangent line to the curve \(y=x^{2}-\) \(3 x+1\) at the point \((2,-1)\)

Differentiate $$ f(x)=\sqrt{a x^{2}-2} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

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

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

Differentiate $$ f(x)=(a-x)(a+x) $$ with respect to \(x\). Assume that \(a\) is a positive constant.