Iterated derivative

Summary

The iterated derivative $D^k f$ of a map $f$ is the derivative of the derivative, taken $k$ times. The Fréchet derivative of $D^{k-1} f$ at a point is a linear map with values in $(k-1)$-linear maps, which is the same as a $k$-linear map. So $D^k f(a)$ is a $k$-linear map, and for a function on $\mathbb{R}^n$ its values on basis vectors are the partial derivatives of order $k$.