The directional derivative of a differentiable multivariable function in the direction of a vector is the instantaneous rate of change of the function while moving in the direction of . In physical terms, this can be thought of as the rate of change of the function while moving with velocity . In many cases, only the directional derivative with respect to space (not time) is desired, making velocity irrelevant; in such a case, we would use the unit vector . In two dimensions, the directional derivative is equal to the dot product of the gradient and directional vector.
Generalized to dimensions, this becomes