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.
The directional derivative of a differentiable function along a unit vector is the function defined by the limit
It can be written in terms of the gradient of f by
where denotes the dot product (Euclidean inner product). At any point p, the directional derivative of f intuitively represents the rate of change in f in the direction of at the point p.
A vector field at a point p naturally gives rise to linear functionals defined on p by evaluating the directional derivative of a differentiable function f along the unit vector where is the vector of the tangent space at p assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at p in the direction of .