Derivatives in Function Analysis - Derivatives of Higher Order

A function sometimes has a class.
It's usually denoted by a function being labeled as
This is read as "n-times differentiable"

Derivative order notation

A n-th order derivative means that the function has been differentiated n-times. So, we have taken the derivative of the function and its subsequent derivatives n times.

In the "prime" notation, you either add an extra ' symbol or indicate it by a number enclosed by parentheses, for example:

But what does this mean exactly?

This means that we can take the derivative of a derivative of a function that many times.

Think of this function as an example:
How many times can you differentiate it?

This will be useful, as subsequent derivatives can give us a lot more information about the behaviour of a function.

The Second Derivative - Inflection Points