# Differentiable functions/R/Higher derivatives/Introduction/Section

The derivative ${\displaystyle {}f'}$ of a differentiable function is also called the first derivative of ${\displaystyle {}f}$. The zeroth derivative is the function itself. Higher derivatives are defined recursively.

## Definition

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

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

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

${\displaystyle {}f^{(n)}(x):=(f^{(n-1)})'(x)\,}$
is called the ${\displaystyle {}n}$-th derivative of ${\displaystyle {}f}$.

The second derivative is written as ${\displaystyle {}f^{\prime \prime }}$, the third derivative as ${\displaystyle {}f^{\prime \prime \prime }}$. If a function is ${\displaystyle {}n}$-times differentiable, then we say that the derivatives exist up to order ${\displaystyle {}n}$. A function ${\displaystyle {}f}$ is called infinitely often differentiable, if it is ${\displaystyle {}n}$-times differentiable for every ${\displaystyle {}n}$.

A differentiable function is continuous due to fact, but its derivative is not necessarily so. Therefore, the following concept is justified.

## Definition

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

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

be a function. The function ${\displaystyle {}f}$ is called continuously differentiable, if ${\displaystyle {}f}$ is differentiable and its derivative ${\displaystyle {}f'}$ is

continuous.

A function is called ${\displaystyle {}n}$-times continuously differentiable, if it is ${\displaystyle {}n}$-times differentiable, and its ${\displaystyle {}n}$-th derivative is continuous.