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. \(\sqrt{65} ;\) let \(f(x)=\sqrt{x}, a=64\), and \(x=65\)

Find the derivative with respect to the independent variable. $$ f(x)=2 \sin x-\cos x $$

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

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. $$ f(x)=4 x^{3}-7 x+1 $$

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. $$ \text { is the length of a fish and } x \text { is the age of the fish. } $$

Find the first and the second derivatives of each function. $$ f(x)=x^{3}-3 x^{2}+1 $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ x^{2}+y^{2}=4 $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=(x-3)^{2} $$

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. \(\sqrt{35} ;\) let \(f(x)=\sqrt{x}, a=36\), and \(x=35\)

Find the derivative at the indicated point from the graph of \(y=f(x)\). \(f(x)=-3 x ; x=-2\)

Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.14) to find the derivative of the inverse. $$ f(x)=\sqrt{x+1}, x \geq-1 $$

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

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

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. $$ y \text { is the number of cells in a petri dish, } x \text { is time. } $$

Find the first and the second derivatives of each function. $$ f(x)=(2 x+4)^{3} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ y=x^{2}+3 y x $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=(4 x+5)^{3} $$

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. \(\sqrt[3]{124}\)

Find the derivative at the indicated point from the graph of \(y=f(x)\). \(f(x)=2 x-3 ; x=-1\)

Find the derivative with respect to the independent variable. $$ f(x)=3 \sin x+5 \cos x $$

Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.14) to find the derivative of the inverse. $$ f(x)=2 x^{2}+2, x \geq 0 $$

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

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. $$ v \text { is the heart rate of a mammal, } x \text { is the mammal's body mass. } $$

Find the first and the second derivatives of each function. $$ g(x)=\frac{1}{x+1} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ x^{3 / 4}+y^{3 / 4}=1 $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\left(1-3 x^{2}\right)^{4} $$

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 \(y=f(x)\). \(f(x)=-5 x+1 ; x=0\)

Find the derivative with respect to the independent variable. $$ f(x)=-\sin x+\cos x $$

Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.14) to find the derivative of the inverse. $$ f(x)=3 x^{2}+2, x \geq 0 $$

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

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. \(y\) is the amount of chemical produced by a chemical reaction, \(x\) is the amount of catalyst added.

Find the first and the second derivatives of each function. $$ h(s)=\frac{1}{s^{2}+2} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ x y-y^{3}=1 $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\left(x^{2}-3 x\right)^{3} $$

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

Find the derivative with respect to the independent variable. $$ f(x)=\cos (x+1) $$

Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.14) to find the derivative of the inverse. $$ f(x)=3-2 x^{3}, x \geq 0 $$

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

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words. 5\. \(y\) is the number of cars leaving a freeway in one minute, \(x\) is the number of cars on the freeway.

Find the first and the second derivatives of each function. $$ g(t)=\sqrt{3 t^{3}+2 t} $$

In Problems \(1-8\), find \(\frac{d y}{d x}\) by implicit differentiation. $$ \sqrt{x y}=x^{2}+1 $$