# Definition: Higher Order Directional Derivative

Let $$(V,\| \|)$$ and $$(W,\| \|)$$ be finitely dimensional normed vector spaces, $$G\subseteq V$$ an open subset, and $$f\colon G\rightarrow W$$ a function. Furthermore, let $$x\in G$$ be a point and $$v_{1},\ldots ,v_{n}$$ be fixed vectors in $$V$$ in a fixed order $$1,2,\ldots,n$$. Then we call the higher order directional derivative of $$f$$ along $$v_{1},\ldots ,v_{n}$$, if

• the higher order directional derivative of $$f$$ along $$v_{1},\ldots ,v_{n-1}$$ exists and if
• its directional derivative along $$v_{n}$$ exists.

We also denote the higher order directional derivative as

$D_{v_{n}}(...D_{v_{2}}(D_{v_{1}}f(x)))$

(read "$$n$$-order directional derivative of $$f$$ along the vectors $$v_{1},\ldots ,v_{n}$$ at the point $$x$$").

Please note that the order of vectors, along which we take the higher order directional derivative is fixed.

Definitions: 1
Proofs: 2

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck