Real function/Higher derivative/Recursive/Definition

Higher derivatives

Let ${}I\subseteq \mathbb {R}$ denote an interval, and let

$f\colon I\longrightarrow \mathbb {R}$

be a function. The function ${}f$ is called ${}n$-times differentiable, if it is ${}(n-1)$-times differentiable, and the ${}(n-1)$-th derivative, that is ${}f^{(n-1)}$, is also differentiable. The derivative

${}f^{(n)}(x):=(f^{(n-1)})'(x)\,$

is called the ${}n$-th derivative of ${}f$.