Definition:Derivative/Higher Derivatives/Higher Order

Definition

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

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

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

Notation

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ as being ridiculously archaic.

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

Also see

• Definition:Differentiability Class

• Definition:Order of Derivative

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Higher Derivatives: $13.45$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.2$

"... and so on."