---
tag: 0F32S73FJN
kind: concept
name: Iterated derivative
aliases: [higher-frechet-derivative, kth-derivative]
msc: [26B05]
uses: [06EH97V2A7]
sources:
  - url: https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative
    locator: Higher derivatives
---

## Summary

The iterated derivative $D^k f$ of a map $f$ is the derivative of the derivative, taken $k$ times.
The [[Fréchet derivative|06EH97V2A7]] 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 $k$.