From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# Derivative of a Product of Functions Exercises

### Example 1

Let f(x) = x + 1 and g(x) = x

• Find f ' (x).

### Example 2

Let f(x) = x + 1 and g(x) = x.

• Find g ' (x).

### Example 3

Let f(x) = x + 1 and g(x) = x.

• Find f ' (x) × g ' (x).

### Example 4

Let f(x) = x + 1 and g(x) = x.

• Find (f × g)'(x).

### Example 5

Let f(x) = x + 1 and g(x) = x.

• Must the derivative of the function f × g be equal to the product of f ' and g '?

### Example 6

Find the derivative of the following function.

• f(x) = x sin x

### Example 7

What's the derivative of f(x)?

• f(x) = excos x

### Example 8

What's the derivative of the following function?

• f(x) = x ln x

### Example 9

What's g ' (x) for the following function?

• g(x) = 5xex

### Example 10

What's the derivative of g(x)?

•  g(x) = (logx)(logx)

### Example 11

What's the derivative of g(x)?

• g(x) = 5ex

### Example 12

Find the derivative of the following function.

• h(x) = (x2 + 2x)ln x

### Example 13

What is h ' (x) for the following function?

• h(x) = ln cos x

### Example 14

Find the derivative of h(x).

### Example 15

Find the derivative of j(x).

• j(x) = ln x + cos x

### Example 16

What is the derivative of f(x) = (x2 + 2)(x3 – 4).

• Use the product rule.

### Example 17

What's the derivative of f(x) = (x2 + 2)(x3– 4)?

• Rewrite f by multiplying the factors together, then take the derivative.

### Example 18

• What's the derivative of

f(x) = x2exsin x

thinking of the function as

f(x) = (x2)(exsin x).

### Example 19

Find the derivative of the function

f(x) = x3sin cos x

in two different ways. Give the answer with everything multiplied out (instead of factoring out common factors).