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 (x^{3})
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) = 4^{7x}
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) = (3x^{2} + 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) = (x^{5} + 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) = e^{sin x}
Example 10
What's the derivative of the following function?
- h(x) = ln x^{3}
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).