# Definition:Directional Derivative

## Definition

Let:

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

be a real-valued function such that the gradient:

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

exists.

Let:

$\mathbf u$

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

 $\displaystyle D_{ \mathbf {u} } f\left({\mathbf x}\right)$ $=$ $\displaystyle \nabla f\left({\mathbf x}\right) \bullet \mathbf u$

where $\bullet$ represents the dot product.