Let $ p\in \mathbb{R^n}$ and $S(p,1)$ be the unit sphere, prove that $df\restriction_{S(p,1)}:S(p,1)\mapsto \mathbb{R}$ is maximized in the direction $\frac{\nabla f}{|| \nabla f||}$
$df(v)=\nabla f\cdot v =||\nabla f ||||v|||cos(\theta )$, now because $v\in S(p,1)\Rightarrow ||v||=1$ and we get $df(v)=||\nabla f ||cos(\theta ) $ which of course maximized when $cos(\theta)=1$
In order to make $\nabla f$ a unit vector, we divide with its length.
is my solution correct ?