Directional derivative

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In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point.^{[citation needed]}

The directional derivative of a multivariable differentiable scalar function along a given vector v at a given point x represents the instantaneous rate of change of the function in the direction v through x.

Many mathematical texts assume that the directional vector is normalized (a unit vector), meaning that its magnitude is equivalent to one. This is by convention and not required for proper calculation. In order to adjust a formula for the directional derivative to work for any vector, one must divide the expression by the magnitude of the vector. Normalized vectors are denoted with a circumflex (hat) symbol: ${\displaystyle \mathbf {\widehat {}} }$.

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

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. The directional derivative is a special case of the Gateaux derivative.