Does gradient gives the rate of change for a field in general in space or at a particular point?

Question

When we want to find the directional derivative in a particular direction then we take grad(f).u, where u-unit vector in the direction we want to find the rate of change or directional derivative. So from the above equation it means the that grad(f) gives the rate of change at a particular point and grad(f).u this gives the component along u.