Gradient Points to the direction of Steepest Slope

Lemma

The gradient gives the direction of the largest directional derivative, hence it points to the direction of the steepest slope.

Proof

The Directional Derivative of a vector function is given by.

$$\begin{array}{rl}\frac{\partial }{\partial \vec{n}}u(\vec{x})& =\vec{n}\cdot \nabla u(\vec{x})\\ & =\Vert \vec{n}{\Vert }_2\cdot \Vert \nabla u(\vec{x}){\Vert }_2\cdot \cos \theta \end{array}$$

where $\theta$ is the angle between the two vectors. i.e $\cos \angle (\nabla u,\mathbf{n})$ The largest directional derivative is given by

$$\begin{array}{rl}\max \left(\frac{\partial }{\partial \vec{n}}u(\vec{x})\right)& =\max \left(\Vert \vec{n}{\Vert }_2\cdot \Vert \nabla u(\vec{x}){\Vert }_2\cdot \cos \theta \right)\\ & =1\cdot \Vert \nabla u(\vec{x}){\Vert }_2\cdot 1\\ & =\Vert \nabla u(\vec{x}){\Vert }_2\end{array}$$

RHS is maximum when $\cos \theta =1,\Rightarrow \theta =0$. This means that the angle between the Direction of derivative and the Gradient is 0 . $\therefore$ Thus the direction of the gradient is the same as the direction of the steepest slope (Because of the maximizing function)