Definition:Directional Derivative

Definition
Let:


 * $f: \R^n \to \R, \mathbf x \mapsto \map f {\mathbf x}$

be a real-valued function such that the gradient:


 * $\nabla \map f {\mathbf x}$

exists.

Let:


 * $\mathbf u$

be a unit vector in $\R^n$.

The directional derivative of $f$ in the direction of $\mathbf u$ is defined as:

where $\cdot$ denotes the dot product.