Directional derivative

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

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

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).