Because that is theOriginally Posted byAMRgradientof a diffrenciable function.

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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.

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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.