Chapter 4

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. $$ \sqrt{65} \text { ; let } f(x)=\sqrt{x}, a=64 \text { , and } x=65 $$

Find the derivative at the indicated point from the graph of each function. $$ f(x)=5 ; x=1 $$

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

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

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

In Problems 1-16, use the product rule to find the derivative with respect to the independent variable. $$ f(x)=(x+5)\left(x^{2}-3\right) $$

Differentiate the functions given in Problems 1-22 with respect to the independent variable. $$ f(x)=4 x^{3}-7 x+1 $$

Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.12) to find the derivative of the inverse. $$ f(x)=\sqrt{2 x+1}, x \geq-\frac{1}{2} $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. $$ \sqrt{35} ; \text { let } f(x)=\sqrt{x}, a=36 \text { , and } x=35 $$

Find the derivative at the indicated point from the graph of each function. $$ f(x)=-3 x ; x=-2 $$

Differentiate the functions with respect to the independent variable. \(f(x)=(4 x+5)^{3}\)

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

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\left(2 x^{3}-1\right)\left(3+2 x^{2}\right) $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=-3 x^{4}+5 x^{2} $$

Find the derivative at the indicated point from the graph of each function. $$ f(x)=4 x-3 ; x=-1 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\left(1-3 x^{2}\right)^{4}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=3 \sin x+5 \cos x-2 \sec x $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=4 e^{1-3 x} $$

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\left(3 x^{4}-5\right)\left(2 x-5 x^{3}\right) $$

Differentiate the functions given in Problems with respect to the independent variable.$$ f(x)=-2 x^{5}+7 x-4 $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. $$ (7.9)^{3} $$

Find the derivative at the indicated point from the graph of each function. $$ f(x)=-5 x+1 ; x=0 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\left(5 x^{2}-3 x\right)^{2}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-\sin x+\cos x-3 \csc x $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=3 e^{2-5 x} $$

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\left(3 x^{4}-x^{2}+1\right)\left(2 x^{2}-5 x^{3}\right) $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=-3 x^{4}+6 x^{2}-2 $$

Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.12) to find the derivative of the inverse. $$ f(x)=3 x^{2}+2, x \geq 0 $$

Find the derivative at the indicated point from the graph of each function. $$ f(x)=2 x^{2} ; x=0 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt{x^{2}+3}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\tan x-\cot x $$

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

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\left(\frac{1}{2} x^{2}-1\right)\left(2 x+3 x^{2}\right) $$

Differentiate the functions given in Problems with respect to the independent variable. $$ f(x)=3-4 x-5 x^{2} $$

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

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=e^{4 x^{2}-2 x+1} $$

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=2\left(3 x^{2}-2 x^{3}\right)\left(1-5 x^{2}\right) $$

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

Find the derivative at the indicated point from the graph of each function. $$ f(x)=(x+2)^{2} ; x=1 $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. $$ \sin \left(\frac{\pi}{2}+0.02\right) $$

Find the derivative at the indicated point from the graph of each function.. $$ f(x)=\cos x ; x=0 $$

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt{3-x^{3}}\)

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

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=e^{7\left(x^{2}+1\right)^{2}} $$

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{5}\left(x^{2}-1\right)\left(x^{2}+1\right) $$

Differentiate the functions given in Problems with respect to the independent variable. $$ g(s)=5 s^{7}+2 s^{3}-5 s $$

Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a calculator. $$ \cos \left(\frac{\pi}{4}-0.01\right) $$

Find the derivative at the indicated point from the graph of each function. $$ f(x)=\sin x ; x=\frac{\pi}{2} $$

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt{5 x+3 x^{4}}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\cos (-5 x) $$