Why the gradient vector gives the direction of maximum increase of a function? In context of multivariable functions, $$f: \Bbb{R}^2\to\Bbb{R} $$
Y know that the gradient vector is defined as $$\nabla f(x,y) = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)$$
And I understand that the partial derivatives gives the increase value in the directions of i and j versor respectively. But, why the gradient vector, compound of these two values gives the direction of maximum increase? Why can't be another vector or direction which gives that? Thank you.
The directional derivative $D_v = \nabla f_p \cdot v = \|\nabla f_p\| \cos \theta_{\nabla f_p,v}$ and since $-1 \leq \cos t \leq 1$ then the derivative is maximal with value $\|\nabla f_p\|$ i.e in the direction of the gradient.