Let $U$ be an open set in and is a differentiable function. If $u\in U$ and $v\in \sR^n$ , then the directional derivative of $f$ in the direction of $v$ is $$ (D_v f)(u) = \frac{d}{ds} f(u+sv) \Big|_{s=0}. $$ In other words, $(D_v f)(u)$ measures how $f$ changes in the direction of $v$ from $u$ .

Alternatively, \begin{eqnarray*} (D_v f)(u) &=& \lim_{h\to 0} \frac{ f(u+ hv) - f(u)}{h} \\ &=& Df(u)\cdot v, \end{eqnarray*}where $Df$ is the Jacobian matrix of $f$ .

Properties

1. $D_v f$ is linear in $v$ . If and , then $$ D_{\lambda v+\mu w}f(u) = \lambda D_{v}f(u) +\mu D_{w}f(u). $$ In particular, $D_0 f=0$ .
2. If $f$ is twice differentiable and , then \begin{eqnarray*} D_v D_w f(u) &=& \frac{\partial^2}{\partial s\partial t} f(u+sv + tw) \Big|_{s=0}, \\ &=& v^T\cdot \operatorname{Hess}f(u)\cdot w, \end{eqnarray*}where $\operatorname{Hess}$ is the Hessian matrix of $f$ .

Example