Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>

c. (f^-1)'(f(2))

Computing the derivative of f(x) = e^-x

c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Let f(x) = sin x. What is the value of f′(π)?

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

a. $[0, 2]$

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

b. At what rate is the volume of the water increasing if the water level is rising at 1/4ft/min.

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

Use differentiation to verify each equation.

d/dx(x / √1−x²) = 1 / (1−x²)^3/2.

At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.

a. Find an equation relating A to w.

An equation​​ of the line tangent to the graph of f at the point (2,7) is y = 4x−1. Find f(2) and f′(2).

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

f(x) = x(x-1)

The volume V of a sphere of radius r changes over time t.

b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?

The volume V of a sphere of radius r changes over time t.

c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?

An equation of the line tangent to the graph of g at x = 3 is y = 5x + 4. Find g(3) and g′(3).

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

g(t) = (t + 1)(t² - t + 1)

If h(1) = 2 and h′(1) = 3, find an equation of the line tangent to the graph of h at x = 1.

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

f(x) = (x - 1)(3x + 4)

If f′(−2) = 7, find an equation of the line tangent to the graph of f at the point (−2,4).

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

h(z) = (z³ + 4z² + z)(z - 1)

Let F(x) = f(x) + g(x),G(x) = f(x) - g(x), and H(x) = 3f(x) + 2g(x), where the graphs of f and g are shown in the figure. Find each of the following.

<IMAGE>

H'(2)

Use the table to find the following derivatives.

<IMAGE>

d/dx (f(x) + g(x)) ∣x=1

Shrinking square The sides of a square decrease in length at a rate of 1 m/s.

a. At what rate is the area of the square changing when the sides are 5 m long?

The sides of a square decrease in length at a rate of 1 m/s.

b. At what rate are the lengths of the diagonals of the square changing?

What is the derivative of y = e^kx?

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

y = x² - a² / x-a, where a is a constant

The legs of an isosceles right triangle increase in length at a rate of 2 m/s.

c. At what rate is the length of the hypotenuse changing?

Find f′(x) if f(x) = 15e^3x.

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

y = x² - 2ax +a² / x-a, where a is a constant

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = (3x+7)¹⁰

Use definition (1) (p. ​133​) to find the slope of the line tangent to the graph of f at P.

f(x) = x² - 5; P(3,4)

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the velocity and acceleration of the object at t = 1. 

f(t) = t² − 4t; 0 ≤ t ≤ 5