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Directional derivative
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<p>In <a href="/facts/Multivariable_calculus/PEs5SgUF">multivariable calculus</a>, the directional derivative measures the rate at which a function changes in a particular direction at a given point. </p><p>The directional derivative of a multivariable <a href="/facts/Differentiable_function/jQqTgLk1">differentiable (scalar) function</a> along a given <a href="/facts/Vector_(mathematics)/U8mM7A5F">vector</a> 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. </p><p>The directional derivative of a <a href="/facts/Scalar_field/8MtPTGob">scalar function</a> <i>f</i> with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: ∇ v f ( x ) = f v ′ ( x ) = D v f ( x ) = D f ( x ) ( v ) = ∂ v f ( x ) = v ⋅ ∇ f ( x ) = v ⋅ ∂ f ( x ) ∂ x . {\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} )\\&=\mathbf {v} \cdot {\nabla f(\mathbf {x} )}\\&=\mathbf {v} \cdot {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {x} }}.\\\end{aligned}}} </p><p>It therefore generalizes the notion of a <a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a>, in which the rate of change is taken along one of the <a href="/facts/Curvilinear_coordinates/RQD7Cajd">curvilinear</a> <a href="/facts/Coordinate_curves/Mtd0q4SP">coordinate curves</a>, all other coordinates being constant. The directional derivative is a special case of the <a href="/facts/Gateaux_derivative/dqdtTtpf">Gateaux derivative</a>. </p>