Df(x) is, for each x, a vector (or can be represented by such a vector, depending upon exactly how you are defining Df(x)) pointing in the direction of fastest increase. Show that if f is differentiable at a and Df(a) is not 0 then moving a short distance in the direction of Df(a) will give a higher value of f and in the opposite direction will give a lower value. You can use the mean value theorem for that.