3.6 Calculating Higher-Order Derivatives

The term “higher-order derivative” may seem intimidating at first.  However, this simply means that we can take the first derivative, second derivative, third derivative, and so on of a function.

This is useful because graphically and analytically, first and second derivatives can provide us with different information. 

The first derivative of a function tells us where f(x) has a relative minimum or maximum because the slope, f’(x), is equal to zero.  

The second derivative of a function provides us with information about the concavity of a function by allowing us to find points of inflection, the location where f(x) changes from concave up to concave down and vice versa.

Ultimately taking the second derivative and subsequent derivatives follow the same process as taking the first derivative of f(x).  With the second derivative, however, we will take the derivative of f’(x) instead.

For example, let’s take the function:

Now that you understand how to calculate higher-order derivatives, practice finding the second derivative of the following practice problems!

Hopefully, this study guide helped you out.  Remember that the one limit that never exists is the limit of your potential in calculus!  Best of luck on your next test and your exams in May.