Chapter 2

Find \(d y / d x\) by implicit differentiation. \(\sin x+2 \cos 2 y=1\)

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

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

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(x)=x+11 $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=7 $$

Area The included angle of the two sides of constant equal length \(s\) of an isosceles triangle is \(\theta\) . (a) Show that the area of the triangle is given by \(A=\frac{1}{2} s^{2} \sin \theta .\) (b) the angle \(\theta\) is increasing at the rate of \(\frac{1}{2}\) radian per minute. Find the rates of change of the area when \(\theta=\pi / 6\) and \(\theta=\pi / 3 .\) (c) Explain why the rate of change of the area of the triangle is not constant even though \(d \theta / d t\) is constant.

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

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

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

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(x)=6 x+3 $$

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

Volume The radius \(r\) of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rates of change of the volume when \(r=9\) inches and \(r=36\) inches. (b) Explain why the rate of change of the volume of the sphere is not constant even though \(d r / d t\) is constant.

Find \(d y / d x\) by implicit differentiation. \(\sin x=x(1+\tan y)\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\sqrt[3]{6 x^{2}+1} $$

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=\left(x^{3}+4 x\right)\left(3 x^{2}+2 x-5\right) \quad c=0 $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(t)=-2 t^{2}+3 t-6 $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=-10 x $$

A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters (b) 60 centimeters?

Find \(d y / d x\) by implicit differentiation. \(\cot y=x-y\)

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

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ y=\left(x^{2}-3 x+2\right)\left(x^{3}+1\right) \quad c=2 $$

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

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=7 x-3 $$

All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is (a) 2 centimeters (b) 10 centimeters?

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

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=2 \sqrt[4]{9-x^{2}} $$

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=\frac{x^{2}-4}{x-3} \quad c=1 $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(x)=x^{2}+4 x^{3} $$

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

All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the surface area changing when each edge is (a) 2 centimeters (b) 10 centimeters?

Find \(d y / d x\) by implicit differentiation. \(x=\sec \frac{1}{y}\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(x)=\sqrt[3]{12 x-5} $$

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=\frac{x-4}{x+4} \quad c=3 $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=4 x-3 x^{3} $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=5-\frac{2}{3} x $$

Volume At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high? (Hint: The formula for the volume of a cone is \(V=\frac{1}{3} \pi r^{2} h . )\)

(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). \(x^{2}+y^{2}=64\)

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

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=x \cos x \quad c=\frac{\pi}{4} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ s(t)=t^{3}+5 t^{2}-3 t+8 $$

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

A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. Water is flowing into the tank at a rate of 10 cubic feet per minute. Find the rate of change of the depth of the water when the water is 8 feet deep.

(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). \(25 x^{2}+36 y^{2}=300\)

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ s(t)=\frac{1}{4-5 t-t^{2}} $$

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=\frac{\sin x}{x} \quad c=\frac{\pi}{6} $$

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=2 x^{3}+6 x^{2}-1 $$

Finding the Derivative by the Limit Process In Exercises \(11-24,\) find the derivative of the function by the limit process. $$ f(x)=x^{2}-5 $$

(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). \(16 y^{2}-x^{2}=16\)

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

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