Higher Order Derivatives

Why

The second derivative (if it exists) is the derivative of the derivative of a function. Can we continue in this way?

Definition

Let $A \subset \R $. Let $f: A \to \R $ be twice differentiable. We call $f$ three times differentiable (or thrice differentiable) if its second derivative is differentiable. We call the derivative of the second derivative of $f$ the third derivative of $f$.

For $n \geq 3$, we call $f$ $n+1$-times differentiable if $f$ is $n$-times differentiable. The $n+1$th derivative of a $n+1$-times differenetiable function is the derivative the $n$th derivative of the function.

Notation

The $n$th derivative of a function $f: A \to \R $ is sometimes denoted $f^{(n)}: A \to \R $.