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Computing Derivatives

Computing Derivatives

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).