1. Higher Order DerivativesBy
Dr. Julia Arnold and Ms. Karen Overman
using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences text

2. Since the derivative of a function is itself a function, it makes sense to think about taking the derivative of the derivative function which we will call a higher order derivative.
The function f” is called the second derivative of f and is the derivative of f’.
If it is possible to continue, then we can consider a third derivative f’’’ or a fourth derivative f(4), a fifth derivative f(5), and on and on …

3. Notation can be one of the following:

4. Example 1: Find the fifth derivative of
There is no short cut to finding the fifth derivative. You have to go through each one until you get to the fifth.

5. Example 2: Find the 2nd derivative of

6. Example 3: Find the second derivative of
Notice you will need to use the Chain Rule to find the first derivative.
Always simplify the derivative before going on to the next derivative.

7. Now take the derivative of f’ to get the second derivative or f’’.
Notice you will need to use the Product Rule to find the derivative
and then use the Chain Rule when you take the derivative of the
second factor.