Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. 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 Gâteaux derivative.

The directional derivative of a scalar function

along a vector

is the function defined by the limit

In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. Without the restriction, this definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.

If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has

where the ${\displaystyle \nabla }$ on the right denotes the gradient and ${\displaystyle \cdot }$ is the dot product. Intuitively, the directional derivative of f at a point x represents the rate of change of f with respect to time when moving past x at velocity v.

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