Directional derivative

A directional derivative of a function $f:U\to\mathbb R$ , where $U\subseteq\mathbb R^n$ is open, at $a\in U$ in the direction $v\in\mathbb R^n$ is the limit

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

when it exists.

If $f$ is differentiable at $a$ , then $D_vf(a)$ exists for every $v$ and equals $Df(a)v$ , where $Df(a)$ is the Fréchet derivative.

Examples

If $f(x,y)=x^2+y^2$ , then at $a=(1,0)$ in the direction $v=(1,1)$ , $D_v f(a)=\lim_{t\to 0}\frac{(1+t)^2+t^2-1}{t}=2.$
For $f(x,y)=|x|$ , one has $D_{(0,1)}f(0,0)=0$ , but $D_{(1,0)}f(0,0)$ does not exist. Its one-sided directional derivative with $t\to0^+$ equals $1$ .