Definition:Directional Derivative

Definition
Let:


 * $f: \R^n \to \R, \mathbf x \mapsto f\left({\mathbf x}\right)$

be a real function of several variables such that the gradient:


 * $\nabla f\left({\mathbf x}\right)$

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 $\bullet$ represents the dot product.