# Definition:Derivative/Higher Derivatives/Higher Order

## Definition

The $n$th derivative of a function $y = \map f x$ is defined as:

$\map {f^{\paren n} } x = \dfrac {\d^n y} {\d x^n} := \begin {cases} \map {\dfrac \d {\d x} } {\dfrac {\d^{n - 1} y} {\d x^{n - 1} } } & : n > 0 \\ y & : n = 0 \end {cases}$

assuming appropriate differentiability for a given $f^{\paren {n - 1} }$.

## Also denoted as

The $n$th derivative of $\map f x$ can variously be denoted as:

$D^n \map f x$
$D_{\map x n} \map f x$
$\dfrac {\d^n} {\d x^n} \map f x$

The $n$ in $f^{\paren n}$ is sometimes written as a roman numeral, but this is considered on this website as being ridiculously archaic.

If the $n$th derivative exists for a function, then $f$ is described as being $n$ times differentiable.

"... and so on."