How do we prove that the gradient points in the direction of the maximum increase? Would it be enough to simply state that the gradient is just the derivates of a function w.r.t all the variables a function depends upon. Since the derivative of a term w.r.t a certain variable gives the maximum increase of that term and since in gradient not only do we account for all possible terms and all the variables upon which a term depends upon but also includes the direction, so it is logical to conclude that the gradient points in the direction of the maximum increase. Note: I do hope that the above proof is adequate but somehow feel that the proof has to be mathematical. Could someone tell me whether or not my intuition is correct?