Editing Directional derivative (section)

{{Short description|Instantaneous rate of change of the function}}
{{refimprove section|date=October 2012|talk=Verifiability of definition}}
{{Calculus |Vector}}

In [[multivariable calculus]], the '''directional derivative''' measures the rate at which a function changes in a particular direction at a given point.{{cn|date=November 2023}}

The directional derivative of a multivariable [[differentiable function|differentiable (scalar) function]] along a given [[vector (mathematics)|vector]] '''v''' at a given point '''x''' intuitively represents the instantaneous rate of change of the function, moving through '''x''' with a direction specified by '''v'''.  

The directional derivative of a [[Scalar field|scalar function]] ''f'' with respect to a vector '''v''' at a point (e.g., position) '''x''' may be denoted by any of the following:
<math display="block">
\begin{aligned}
\nabla_{\mathbf{v}}{f}(\mathbf{x})
&=f'_\mathbf{v}(\mathbf{x})\\
&=D_\mathbf{v}f(\mathbf{x})\\
&=Df(\mathbf{x})(\mathbf{v})\\
&=\partial_\mathbf{v}f(\mathbf{x})\\
&=\mathbf{v}\cdot{\nabla f(\mathbf{x})}\\
&=\mathbf{v}\cdot \frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}.\\
\end{aligned}
</math>

It therefore generalizes the notion of a [[partial derivative]], in which the rate of change is taken along one of the [[Curvilinear coordinates|curvilinear]] [[coordinate curves]], all other coordinates being constant.
The directional derivative is a special case of the [[Gateaux derivative]].