Because that is the gradient of a diffrenciable function.Originally Posted by AMR
I can prove it for .
Let at a point.
Let, be any unit vector.
Then, directional derivative of,
The largest possible value of,
Thus, maximum is,
Thus, it is in the in the direction of the gradient. Thus, it is a multiple of the vector, hence the constant of proportionality.
I am thinking about several variables.
I guess the proof is based on finding the critical points. Since the critical points are determined by partial derivatives they appear over here. I will think more about it.