\begin{definition}[Definition:Directional Derivative]
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:
:$\dfrac \partial {\partial \mathbf u} \map f {\mathbf x} = \nabla \map f {\mathbf x} \cdot \mathbf u$
where $\cdot$ denotes the dot product.
\end{definition}