Directional Derivative

Definition

Let $f:\mathcal{D}\subset {\mathbb{R}}^n\to \mathbb{R}$ be a differentiable, a point $\mathbf{p}\in \mathcal{D}$, and a non-zero directional vector $\mathbf{v}\in {\mathbb{R}}^n$. Then, ${\nabla }_{\mathbf{v}}f(\mathbf{p}):={\lim }_{h\to 0}\frac{f(\mathbf{p}+h\mathbf{v})-f(\mathbf{p})}{h}$ ${\nabla }_{\mathbf{v}}f(\mathbf{p})$ is the directional derivative of $f$ at $\mathbf{p}$ with respect to $\mathbf{v}$

If the function $f$ is differentiable at $\mathbf{p}$, the directional derivative can be calculated by Gradient ${\nabla }_{\mathbf{v}}f(\mathbf{p})=\nabla f(\mathbf{p})\cdot \mathbf{v}$ where $\nabla f(\mathbf{p})$ is the Gradient of $f$ at the point $\mathbf{p}$

Facts

Existence of directional derivative does not imply existence of derivative