Chapter 2

Using Related Rates In Exercises \(1-4,\) assume that \(x\) and \(y\) are both differentiable functions of \(t\) and find the required values of \(d y / d t\) and \(d x / d t .\) $$ \begin{array}{rlrl}{y=\sqrt{x}} & {\text { (a) } \frac{d y}{d t} \text { when } x=4} & {} & {\frac{d x}{d t}=3} \\ {} & {\text { (b) } \frac{d x}{d t} \text { when } x=25} & {} & {\frac{d y}{d t}=2}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{2}+y^{2}=9\)

Using the Product Rule In Exercises 1-6, use the Product Rule to find the derivative of the function. $$ g(x)=\left(x^{2}+3\right)\left(x^{2}-4 x\right) $$

Find \(d y / d x\) by implicit differentiation. \(x^{2}-y^{2}=25\)

Using the Product Rule In Exercises 1-6, use the Product Rule to find the derivative of the function. $$ y=(3 x-4)\left(x^{3}+5\right) $$

Using Related Rates In Exercises \(1-4,\) assume that \(x\) and \(y\) are both differentiable functions of \(t\) and find the required values of \(d y / d t\) and \(d x / d t .\) $$ \begin{array}{rlrl}{x y=4} & {\text { (a) } \frac{d y}{d t} \text { when } x=8} & {\frac{d x}{d t}=10} \\ {} & {\text { (b) } \frac{d x}{d t} \text { when } x=1} & {} & {\frac{d y}{d t}=-6}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{1 / 2}+y^{1 / 2}=16\)

Decomposition of a Composite Function In Exercises \(1-6,\) complete the table. $$ \begin{array}{lll}{y=f(g(x))} & {u=g(x)} & {y=f(u)} \\ {y=\sqrt{x^{3}-7}} & {}\end{array} $$

Using the Product Rule In Exercises 1-6, use the Product Rule to find the derivative of the function. $$ h(t)=\sqrt{t}\left(1-t^{2}\right) $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=12 $$

Using Related Rates In Exercises \(1-4,\) assume that \(x\) and \(y\) are both differentiable functions of \(t\) and find the required values of \(d y / d t\) and \(d x / d t .\) $$ \begin{aligned} x^{2}+y^{2}=25 & \text { (a) } \frac{d y}{d t} \text { when } x=3, y=4 \quad \frac{d x}{d t}=8 \\ & \text { (b) } \frac{d x}{d t} \text { when } x=4, y=3 \quad \frac{d y}{d t}=-2 \end{aligned} $$

Find \(d y / d x\) by implicit differentiation. \(2 x^{3}+3 y^{3}=64\)

Decomposition of a Composite Function In Exercises \(1-6,\) complete the table. $$ \begin{array}{lll}{y=f(g(x))} & {u=g(x)} & {y=f(u)} \\ {y=3 \tan \left(\pi x^{2}\right)} & {}\end{array} $$

Using the Product Rule In Exercises 1-6, use the Product Rule to find the derivative of the function. $$ g(s)=\sqrt{s}\left(s^{2}+8\right) $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(x)=-9 $$

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=2 x^{2}+1 ; \frac{d x}{d t}=2 \text { centimeters per second }} \\ {\begin{array}{ll}{\text { (a) } x=-1} & {\text { (b) } x=0} & {\text { (c) } x=1}\end{array}}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{3}-x y+y^{2}=7\)

Using the Product Rule In Exercises 1-6, use the Product Rule to find the derivative of the function. $$ f(x)=x^{3} \cos x $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=x^{7} $$

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ f(x)=3-5 x, \quad(-1,8) $$

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\frac{1}{1+x^{2}} ; \frac{d x}{d t}=6 \text { inches per second }} \\ {\begin{array}{ll}{\text { (a) } x=-2} & {\text { (b) } x=0} & {\text { (c) } x=2}\end{array}}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{2} y+y^{2} x=-2\)

Using the Product Rule In Exercises 1-6, use the Product Rule to find the derivative of the function. $$ g(x)=\sqrt{x} \sin x $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=x^{12} $$

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ g(x)=\frac{3}{2} x+1, \quad(-2,-2) $$

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\tan x ; \frac{d x}{d t}=3 \text { feet per second }} \\\ {\begin{array}{llll}{\text { (a) } x=-\frac{\pi}{3}} & {\text { (b) } x=-\frac{\pi}{4}} & {\text { (c) } x=0}\end{array}}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{3} y^{3}-y=x\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=(4 x-1)^{3} $$

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ f(x)=\frac{x}{x^{2}+1} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=\frac{1}{x^{5}} $$

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ g(x)=x^{2}-9, \quad(2,-5) $$

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\cos x ; \frac{d x}{d t}=4 \text { centimeters per second }} \\ {\begin{array}{llll}{\text { (a) } x=\frac{\pi}{6}} & {\text { (b) } x=\frac{\pi}{4}} & {\text { (c) } x=\frac{\pi}{3}}\end{array}}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(\sqrt{x y}=x^{2} y+1\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=5\left(2-x^{3}\right)^{4} $$

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ g(t)=\frac{3 t^{2}-1}{2 t+5} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=\frac{3}{x^{7}} $$

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ f(x)=5-x^{2}, \quad(3,-4) $$

Related Rates Consider the linear function $$y=a x+b$$ If \(x\) changes at a constant rate, does \(y\) change at a constant rate? If so, does it change at the same rate as \(x ?\) Explain.

Find \(d y / d x\) by implicit differentiation. \(x^{3}-3 x^{2} y+2 x y^{2}=12\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ g(x)=3(4-9 x)^{4} $$

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ h(x)=\frac{\sqrt{x}}{x^{3}+1} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(x)=\sqrt[5]{x} $$

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ f(t)=3 t-t^{2}, \quad(0,0) $$

In your own words, state the guidelines for solving related-rate problems.

Find \(d y / d x\) by implicit differentiation. \(4 \cos x \sin y=1\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(t)=(9 t+2)^{2 / 3} $$

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ f(x)=\frac{x^{2}}{2 \sqrt{x}+1} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(x)=\sqrt[4]{x} $$

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ h(t)=t^{2}+4 t, \quad(1,5) $$

Area The radius \(r\) of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area when (a) \(r=8\) centimeters and (b) \(r=32\) centimeters.