Updated on: 2026-03-06

Directional Derivatives

D_{\mathbf{u}}f(\mathbf{a})=\lim_{h\to 0}\frac{f(\mathbf{a}+h\mathbf{u})-f(\mathbf{a})}{h}

Proposition

Suppose $f:X\mapsto \mathbb{R}$ is differentiable at $a$ . Then for any unit vector $u$ , the directional derivative at $a$ in the direction of $u$ exists, and

D_u f(a)=\nabla f(a)\cdot u

The maximum/minimum value of $D_u f(a)$ is $\plusmn \Vert \nabla f(a) \Vert$