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The gradient of a scalar function is a vector that whose magnitude tells about the maximum rate of change of the function with respect to the distance and its direction tells us the direction of that maximum increase.

Mathematically the gradient is written as ∇f = i*∂f/∂x+ j*∂f/∂y + k*∂f/∂z.

My question is that how this term i*∂f/∂x+ j*∂f/∂y + k*∂f/∂z tells us the direction of maximum increase. What I have understood is that the term i*∂f/∂x+ j*∂f/∂y + k*∂f/∂z should be equal to just df/dl (in some direction) as term i*∂f/∂x+ j*∂f/∂y + k*∂f/∂z just tells us about the ratio of differential change of function "f" to the differential change in length "l" in some direction. So how one can say that the term df/dl (in some direction) = i*∂f/∂x+ j*∂f/∂y + k*∂f/∂z is actually the maximum increase and direction is telling us the direction of that maximum increase?

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Let $v=(v_1,v_2,v_2)$ be any direction. The rate of change of f in this direction at a point $p$ is (chain rule) $$ {d \over dt}_{|t=0} f(p+tv)=\langle \operatorname{grad}f,v\rangle $$ Think of this as a function of $v$ and use the fact, that $\langle z,v\rangle$ is maximal if $z=v$ or minimal if $z=-v$

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