Definition:Derivative/Higher Derivatives/Higher Order

Definition
The $n$th derivative of a function $f \left({x}\right)$ is defined as:
 * $f^{\left({n}\right)} \left({x}\right) = \dfrac {\mathrm d^n} {\mathrm d x^n} := \begin{cases}

\dfrac {\mathrm d} {\mathrm d x} \left({\dfrac {\mathrm d^{n-1}} {\mathrm d x^{n-1} } }\right) & : n > 0 \\ x & : 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

 * Definition:Differentiability Class


 * Definition:Order of Derivative