When "f" is a function from an open subset of R"n" to R"m", then the directional derivative of "f" in a chosen direction is the best linear approximation to "f" at that point and in that direction. But when , no single directional derivative can give a complete picture of the behavior of "f". The total derivative gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds: