Why does the formula for the gradient - that is (for functions of 2 variables), the partial with respect to x plus the partial with respect to y give the direction of greatest increase? i.e. the direction of maximum at some point on a surface is given by [tex]f_xi+f_yj[/tex] And why, when you times each partial derivative with the corresponding components of a vector, it gives the derivative of the surface in the direction of that vector? i.e. the derivative of f(x,y) in the direction of <a, b> is [tex]af_x+bf_y[/tex] The proofs don't offer an understanding of why the formula does it what does - at least for me My lack of understanding of this may have something to do with the fact that I still don't get intuitively why a change in the x direction plus change in y direction gives the total change.