Chapter 2

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=\sqrt[3]{x}(\sqrt{x}+3) $$

Finding an Equation of a Tangent Line In Exercises \(25-32,(\text { a) find an equation of the tangent line to the graph of } f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=\sqrt{x-1}, \quad(5,2) $$

Find the slope of the tangent line to the graph at the given point. Bifolium: \(\left(x^{2}+y^{2}\right)^{2}=4 x^{2} y\) Point: \((1,1)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(v)=\left(\frac{1-2 v}{1+v}\right)^{3} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ h(s)=\left(s^{3}-2\right)^{2} $$

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(x)=\frac{8}{x^{2}} \quad(2,2) $$

Finding an Equation of a Tangent Line In Exercises \(25-32,(\text { a) find an equation of the tangent line to the graph of } f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=x+\frac{4}{x}, \quad(-4,-5) $$

Find the slope of the tangent line to the graph at the given point. Folium of Descartes: \(x^{3}+y^{3}-6 x y=0\) Point: \(\left(\frac{4}{3}, \frac{8}{3}\right)\)

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

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ h(x)=\left(x^{2}+3\right)^{3} $$

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(t)=2-\frac{4}{t} \quad(4,1) $$

Finding an Equation of a Tangent Line In Exercises \(25-32,(\text { a) find an equation of the tangent line to the graph of } f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=\frac{6}{x+2}, \quad(0,3) $$

Evaporation As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area \(\left(S=4 \pi r^{2}\right) .\) Show that the radius of the raindrop decreases at a constant rate.

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

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=\frac{2-\frac{1}{x}}{x-3} $$

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(x)=-\frac{1}{2}+\frac{7}{5} x^{3} \quad\left(0,-\frac{1}{2}\right) $$

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{2}} & {2 x-y+1=0}\end{array}$$

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ g(x)=\left(2+\left(x^{2}+1\right)^{4}\right)^{3} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ g(x)=x^{2}\left(\frac{2}{x}-\frac{1}{x+1}\right) $$

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=2 x^{2}} & {4 x+y+3=0}\end{array}$$

Electricity The combined electrical resistance \(R\) of two resistors \(R_{1}\) and \(R_{2},\) connected in parallel, is given by $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$$ where \(R, R_{1},\) and \(R_{2}\) are measured in ohms. \(R_{1}\) and \(R_{2}\) are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is \(R\) changing when \(R_{1}=50\) ohms and \(R_{2}=75\) ohms?

Finding a Derivative Using Technology In Exercises \(35-40,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. $$ y=\frac{\sqrt{x}+1}{x^{2}+1} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=\left(2 x^{3}+5 x\right)(x-3)(x+2) $$

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{3}} & {3 x-y+1=0}\end{array}$$

Adiabatic Expansion When a certain polyatomic gas undergoes adiabatic expansion, its pressure \(p\) and volume \(V\) satisfy the equation \(p V^{1.3}=k,\) where \(k\) is a constant. Find the relationship between the related rates \(d p / d t\) and \(d V / d t .\)

Finding a Derivative Using Technology In Exercises \(35-40,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. $$ y=\sqrt{\frac{2 x}{x+1}} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=\left(x^{3}-x\right)\left(x^{2}+2\right)\left(x^{2}+x-1\right) $$

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(x)=2(x-4)^{2} \quad(2,8) $$

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{3}+2} & {3 x-y-4=0}\end{array}$$

Roadway Design Cars on a certain roadway travel on a circular arc of radius \(r .\) In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude \(\theta\) from the horizontal (see figure). The banking angle must satisfy the equation \(r g \tan \theta=v^{2},\) where \(v\) is the velocity of the cars and \(g=32\) feet per second per second is the acceleration due to gravity. Find the relationship between the related rates \(d v / d t\) and \(d \theta / d t\) .

Finding a Derivative Using Technology In Exercises \(35-40,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. $$ y=\sqrt{\frac{x+1}{x}} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. \(f(x)=\frac{x^{2}+c^{2}}{x^{2}-c^{2}}, \quad c\) is a constant

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(\theta)=4 \sin \theta-\theta \quad(0,0) $$

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\\ {f(x)=\frac{1}{\sqrt{x}}} & {x+2 y-6=0}\end{array}$$

A balloon rises at a rate of 4 meters per second from a point on the ground 50 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 50 meters above the ground.

Finding a Derivative Using Technology In Exercises \(35-40,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. $$ g(x)=\sqrt{x-1}+\sqrt{x+1} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. \(f(x)=\frac{c^{2}-x^{2}}{c^{2}+x^{2}}, \quad c\) is a constant

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ g(t)=-2 \cos t+5 \quad(\pi, 7) $$

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\\ {f(x)=\frac{1}{\sqrt{x-1}}} & {x+2 y+7=0}\end{array}$$

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ f(t)=t^{2} \sin t $$

Finding a Derivative Using Technology In Exercises \(35-40,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. $$ y=\frac{\cos \pi x+1}{x} $$

In Exercises 39–52, find the derivative of the function. $$ f(x)=x^{2}+5-3 x^{-2} $$

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ f(\theta)=(\theta+1) \cos \theta $$

Finding a Derivative Using Technology In Exercises \(35-40,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. $$ y=x^{2} \tan \frac{1}{x} $$

In Exercises 39–52, find the derivative of the function. $$ f(x)=x^{3}-2 x+3 x^{-3} $$

(a) Use implicit differentiation to find an equation of the tangent line to the ellipse \(\frac{x^{2}}{2}+\frac{y^{2}}{8}=1\) at \((1,2)\). (b) Show that the equation of the tangent line to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) at \(\left(x_{0}, y_{0}\right)\) is \(\frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}=1\).

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ f(t)=\frac{\cos t}{t} $$

In Exercises 39–52, find the derivative of the function. $$ g(t)=t^{2}-\frac{4}{t^{3}} $$

(a) Use implicit differentiation to find an equation of the tangent line to the hyperbola \(\frac{x^{2}}{6}-\frac{y^{2}}{8}=1\) at \((3,-2)\). (b) Show that the equation of the tangent line to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) at \(\left(x_{0}, y_{0}\right)\) is \(\frac{x_{0} x}{a^{2}}-\frac{y_{0} y}{b^{2}}=1\).

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ f(x)=\frac{\sin x}{x^{3}} $$