Directional derivative

In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given point. If the vector is multiplied by a scalar, the corresponding directional derivative is multiplied by the same scalar.

Some elementary texts instead use the phrase "directional derivative in the direction of v" for the rate of change in a function per unit distance in that direction. In that convention the nonzero vector v is first normalized to the unit vector ${\displaystyle {\hat {\mathbf {v} }}=\mathbf {v} /\|\mathbf {v} \|}$, where the normalized vector is denoted with a circumflex (hat) symbol: ${\displaystyle \mathbf {\widehat {}} }$.

The directional derivative of a scalar function f with respect to a vector v may be denoted by any of the following: $${\displaystyle {\begin{aligned}\nabla _{\mathbf {v} }{f}(\mathbf {x} )&=f'_{\mathbf {v} }(\mathbf {x} )\\&=D_{\mathbf {v} }f(\mathbf {x} )\\&=Df(\mathbf {x} )(\mathbf {v} )\\&=\partial _{\mathbf {v} }f(\mathbf {x} )\\&={\frac {\partial f(\mathbf {x} )}{\partial \mathbf {v} }}\\&=\mathbf {v} \cdot {\nabla f(\mathbf {x} )}\\&=\mathbf {v} \cdot {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {x} }}.\\\end{aligned}}}$$

It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. In functional analysis, the analogous notion for functions between topological vector spaces is the Gateaux derivative.