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
\[\left(D_{v}f\right)\left(x\right),\]
(read "directional derivative of \(f\) along the vector \(v\) at the point \(x\)").
Definitions: 1