# Definition:Derivative/Higher Derivatives/Higher Order

< Definition:Derivative | Higher Derivatives(Redirected from Definition:Nth Derivative)

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## Contents

## 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**.

## Also see

## 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."

- 1990: H.A. Priestley:
*Introduction to Complex Analysis*(revised ed.) ... (previous) ... (next): Notation and terminology