Higher derivatives

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 $n>m$.