Gradient as the Direction of Fastest Increase

Theorem

• Suppose that $f:{\mathbb{R}}^n\to \mathbb{R}$ is Differentiable at $x_0$
• Suppose $\nabla f(x_0)\ne 0$
• The unit vector $n=\frac{\nabla f(x_0)}{||\nabla f(x_0)||}$ maximizes $Df(x_0)$

Intuition

$Df(x_0)n=\nabla f(x_0)\ast n=||\nabla f(x_0)||||n||\cos \theta =||\nabla f(x_0)||\cos \theta$ And, $\cos (\theta )$ maximized at $\theta =0$