first and second derivative of abs of dot product

Question

I need your help in solving the following:

Given two vectors $x,p \in \mathbb{R}^n$, and let $p \cdot x$ denote the dot product between $p$ and $x$ what is: $$\frac{\partial}{\partial x} \left| p \cdot x\right| = ?, \frac{\partial^2}{\partial x} \left| p \cdot x\right| = ?$$

Thanks in advance.

Answer 1

Note that $$\frac{d}{dx}|y| = \frac{y}{|y|}\frac{dy}{dx}.$$ Consequently, $$\frac{\partial}{\partial x_i}\left|\sum_{k=1}^n{p_k x_k}\right| = p_i\frac{\sum_{k=1}^n{p_k x_k}}{\left|\sum_{k=1}^n{p_k x_k}\right|}.$$ Can you continue from here?