Differentiable functions/R/Higher derivatives/Introduction/Section

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

Definition

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 ${}f$.

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

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

Definition

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

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

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

continuous.

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