# Definition:Directional Derivative

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## 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:

- $\mathbf u$

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

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.

## Sources

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 13.6$