Higher derivatives - Imported Knowlpedia Content

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.

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 $f(x)=x^m$ (a polynomial), $f^{(n)}\equiv 0$ for all $n>m$ .