# Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given unit vector intuitively represents the rate of change of the function in the direction of that vector. It therefore generalizes the notion of a partial derivative in which the direction is always taken along one of the coordinate axes.

## Definition

The directional derivative of a differentiable function [itex]f(\vec{x}) = f(x_1, x_2, \ldots, x_n)[itex] along a unit vector [itex]\vec{v} = (v_1, \ldots, v_n)[itex] is the function defined by the limit

[itex]D_{\vec{v}}{f} = \lim_{h \rightarrow 0}{\frac{f(\vec{x} + h\vec{v}) - f(\vec{x})}{h}}[itex]

It can be written in terms of the gradient [itex]\nabla(f)[itex] of [itex]f[itex] by

[itex]D_{\vec{v}}{f} = \nabla(f) \cdot \vec{v}[itex]

where [itex]\cdot[itex] denotes the dot product (Euclidean inner product). At any point [itex]p[itex], the directional derivative of [itex]f[itex] intuitively represents the rate of change in [itex]f[itex] in the direction of [itex]\vec{v}[itex] at the point [itex]p[itex].

## The directional derivative in differential geometry

A vector field at a point [itex]p[itex] naturally gives rise to linear functionals defined on [itex]p[itex] by evaluating the directional derivative of a differentiable function [itex]f[itex] along the unit vector [itex]\vec{v}/||\vec{v}||[itex] where [itex]\vec{v}[itex] is the vector of the tangent space at [itex]p[itex] assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at [itex]p[itex] in the direction of [itex]\vec{v}[itex].

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