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
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.