Definition: 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 \(v\in V\) a fixed vector. Then we call the directional derivative of \(f\) along \(v\), if the limit. \[\operatorname {lim} _{\substack{t\rightarrow 0,\\t\neq 0}}\,{\frac {f(x+tv)-f(x)}{t}}\]

exists. We also denote the directional derivative as


(read "directional derivative of \(f\) along the vector \(v\) at the point \(x\)").

  1. Definition: Higher Order Directional Derivative

Definitions: 1

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Adapted from CC BY-SA 3.0 Sources:

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