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.


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