For the question highlighted below, I've played around with the right side of the equality but am not sure how to express the left side in order to prove the equality.
Thanks in advance
Let $f(x)=|x|$, $x\in \mathbb{R^n}$. Show that if $x,y\neq 0$, then $$D_y f(x)=\dfrac{\langle x,y \rangle}{|x|}$$
Note that $\frac{\partial f}{\partial x_j}(x) = \frac{\partial}{\partial x_j} \sqrt{\sum_{i=1}^n{x_i^2}} = \frac{x_j}{\sqrt{\sum_{i=1}^n{x_i^2}}}$, when $x \neq 0$. This means that $\nabla f(x) = \frac{1}{|x|}x$, for $x \neq 0$. Finally for $y \neq 0$ we have that $D_y f(x) = \langle\nabla f(x),y\rangle = \frac{1}{|x|}\langle x, y \rangle$.