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Concept
Directional derivative
Summary
A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point.
The directional derivative of a multivariable differentiable (scalar) 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.
The directional derivative of a scalar function f with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:
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.
The directional derivative of a scalar function
along a vector
is the function defined by the limit
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 unit vector v at x, and one has
where the on the right denotes the gradient, is the dot product and v is a unit vector. This follows from defining a path and using the definition of the derivative as a limit which can be calculated along this path to get:
Intuitively, the directional derivative of f at a point x represents the rate of change of f, in the direction of v with respect to time, when moving past x.
In a Euclidean space, some authors define the directional derivative to be with respect to an arbitrary nonzero vector v after normalization, thus being independent of its magnitude and depending only on its direction.
This definition gives the rate of increase of f per unit of distance moved in the direction given by v. In this case, one has
or in case f is differentiable at x,
In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector.
Official source
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