# The Chain Rule Exercises

### Example 1

The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.

•  f(x) = cos(sin x)

### Example 2

The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.

•  f(x) = ln (x3)

### Example 3

The function below is the composition of two other functions. Without rewriting the original function, what are the inside and outside functions?

• f(x) = 47x

### Example 4

The function below is the composition of two other functions. Without rewriting the original function, determine the inside and outside functions.

• f(x) = (3x2 + 1)4

### Example 5

The function below is the composition of two other functions. Without rewriting the original function, what are the inside and outside functions?

• f(x) =

### Example 6

Find the derivative of the function using the chain rule.

• h(x) = (x5 + 4)99

### Example 7

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

• h(x) = sin(ln x)

### Example 8

What is h ' (x)?

• h(x) = ln(ln x)

### Example 9

What is the derivative of the following function?

• h(x) = esin x

### Example 10

What's the derivative of the following function?

• h(x) = ln x3

### Example 11

Let h(x) = (ln x)2.
The following equation for h ' (x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the equation.

### Example 12

Let h(x) = (ln x)2.
The following equation for h ' (x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the equation.

• h ' (x) = 2(ln x)

### Example 13

Let h(x) = (ln x)2.
The following equation for h ' (x) comes from applying the chain rule incorrectly. Identify the mistake(s) in the equation.

### Example 14

Let h(x) = (ln x)2.

• Find a correct formula for h ' (x).