Suppose that is a smooth map between smooth manifolds; then the differential of f at a point x is, in some sense, the best linear approximation of f near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at x to the tangent space of N at f(x). Hence it can be used to push tangent vectors on M forward to tangent vectors on N.