Higher derivatives - Knowlpedia

A higher derivative of a function $f$ is a derivative of order $n\ge 2$ , defined recursively by $f^{(0)}=f$ , $f^{(1)}=f'$ , and $f^{(n)}=(f^{(n-1)})'$ wherever the derivative exists.

Applications

Higher derivatives quantify finer local behavior and are central in approximation results such as Taylor's theorem with remainder. They also determine smoothness classes such as class C^k functions.

Examples

For $f(x)=e^x$ , one has $f^{(n)}(x)=e^x$ for every $n\ge 0$ .
For $f(x)=x^m$ , where $m$ is a nonnegative integer, one has $f^{(n)}\equiv 0$ for all $n>m$ .