Directional derivative - Imported Knowlpedia Content

A directional derivative of a function $f:U\to \mathbb{R}$ (with $U\subseteq \mathbb{R}^n$ ) at a point $a\in U$ in the direction $v\in \mathbb{R}^n$ is the one-sided limit

D_v f(a)=\lim_{t\to 0^+}\frac{f(a+tv)-f(a)}{t},

If $f$ is differentiable at $a$ (as a map into $\mathbb{R}$ ), then $D_v f(a)$ exists for every $v$ and equals $Df(a)v$ , where $Df(a)$ is the Fréchet derivative. For scalar functions, directional derivatives are encoded by the gradient.

Examples

If $f(x,y)=x^2+y^2$ , then at $a=(1,0)$ and $v=(1,1)$ , $D_v f(a)=\lim_{t\to 0^+}\frac{(1+t)^2+t^2-1}{t}=2.$
For $f(x,y)=|x|$ , one has $D_{(0,1)}f(0,0)=0$ , while $D_{(1,0)}f(0,0)=1$ (one-sided).

Remarks

when it exists (the limit uses a one-sided limit along the line through $a$ in direction $v$ ).