Chapter 2

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=x^{3}-12 x $$

A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet. (a) Water is being pumped into the trough at 2 cubic feet per minute. How fast is the water level rising when the depth \(h\) is 1 foot? (b) The water is rising at a rate of \(\frac{3}{8}\) inch per minute when \(h=2 .\) Determine the rate at which water is being pumped into the trough.

(a) find two explicit functions by solving the equation for in terms of (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find \(d y / d x\) implicitly and show that the result is equivalent to that of part (c). \(x^{2}+y^{2}-4 x+6 y+9=0\)

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

Using the Constant Multiple Rule In Exercises 19-24, complete the table to find the derivative of the function without using the Quotient Rule. $$ \text {Function} \quad \text {Rewrite} \quad \text {Differentiate} \quad \text {Simplify} $$ \(y=\frac{5 x^{2}-3}{4}\)

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(t)=\pi \cos t $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=x^{3}+x^{2} $$

A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(x y=6, \quad(-6,-1)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{1}{\sqrt{3 x+5}} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=x^{2}-\frac{1}{2} \cos x $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=\frac{1}{x-1} $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(y^{3}-x^{2}=4, \quad(2,2)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ g(t)=\frac{1}{\sqrt{t^{2}-2}} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=7+\sin x $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=\frac{1}{x^{2}} $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(y^{2}=\frac{x^{2}-49}{x^{2}+49}, \quad(7,0)\)

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

Using the Constant Multiple Rule In Exercises 19-24, complete the table to find the derivative of the function without using the Quotient Rule. $$ \text {Function} \quad \text {Rewrite} \quad \text {Differentiate} \quad \text {Simplify} $$ \(y=\frac{4 x^{3 / 2}}{x}\)

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

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=\sqrt{x+4} $$

A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see figure). (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(x^{2 / 3}+y^{2 / 3}=5, \quad(8,1)\)

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

Using the Constant Multiple Rule In Exercises 19-24, complete the table to find the derivative of the function without using the Quotient Rule. $$ \text {Function} \quad \text {Rewrite} \quad \text {Differentiate} \quad \text {Simplify} $$ \(y=\frac{2 x}{x^{1 / 3}}\)

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=\frac{5}{(2 x)^{3}}+2 \cos x $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=\frac{4}{\sqrt{x}} $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \((x+y)^{3}=x^{3}+y^{3}, \quad(-1,1)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=x \sqrt{1-x^{2}} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=\frac{4-3 x-x^{2}}{x^{2}-1} $$

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. $$ f(x)=x^{2}+3, \quad(-1,4) $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(x^{3}+y^{3}=6 x y-1, \quad(2,3)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{1}{2} x^{2} \sqrt{16-x^{2}} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=\frac{x^{2}+5 x+6}{x^{2}-4} $$

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. $$ f(x)=x^{2}+2 x-1, \quad(1,2) $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(\tan (x+y)=x, \quad(0,0)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{x}{\sqrt{x^{2}+1}} $$

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=x\left(1-\frac{4}{x+3}\right) $$

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. $$ f(x)=x^{3}, \quad(2,8) $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. \(x \cos y=1, \quad\left(2, \frac{\pi}{3}\right)\)

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

Finding a Derivative In Exercises \(25-38\) , find the derivative of the algebraic function. $$ f(x)=x^{4}\left(1-\frac{2}{x+1}\right) $$

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. $$ f(x)=x^{3}+1, \quad(-1,0) $$

Find the slope of the tangent line to the graph at the given point. Witch of Agnesi: \(\left(x^{2}+4\right) y=8\) Point: \((2,1)\)

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

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

In Exercises 25–30, complete the table to find the derivative of the function. $$ y=\frac{\sqrt{x}}{x} $$

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. $$ f(x)=\sqrt{x}, \quad(1,1) $$

Find the slope of the tangent line to the graph at the given point. Cissoid: \((4-x) y^{2}=x^{3}\) Point: \((2,2)\)

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