Definition:Derivative/Higher Derivatives/Higher Order

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Definition

The $n$th derivative of a function $y = f \left({x}\right)$ is defined as:

$f^{\left({n}\right)} \left({x}\right) = \dfrac {\mathrm d^n y} {\mathrm d x^n} := \begin{cases} \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac {\mathrm d^{n-1}y} {\mathrm d x^{n-1} } }\right) & : n > 0 \\ y & : n = 0 \end{cases}$

assuming appropriate differentiability for a given $f^{\left({n-1}\right)}$.


Also denoted as

The $n$th derivative of $f \left({x}\right)$ can variously be denoted as:

$D^n f \left({x}\right)$
$D_{x \left({n}\right)} f \left({x}\right)$
$\dfrac{\mathrm d^n}{\mathrm d x^n} f \left({x}\right)$


The $n$ in $f^{\left({n}\right)}$ is sometimes written as a roman numeral, but this is considered on this website as being laughably archaic and ridiculous.


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


Also see


Sources

"... and so on."