Define what it means for a function $f: \mathbb{R}^{a} \rightarrow \mathbb{R}^{b}$ to be differentiable at a point $p \in \mathbb{R}^{a}$ with derivative a linear map $\left.D f\right|_{p} .$

State the Chain Rule for differentiable maps $f: \mathbb{R}^{a} \rightarrow \mathbb{R}^{b}$ and $g: \mathbb{R}^{b} \rightarrow \mathbb{R}^{c}$. Prove the Chain Rule.

Let $\|x\|$ denote the standard Euclidean norm of $x \in \mathbb{R}^{a}$. Find the partial derivatives $\frac{\partial f}{\partial x_{i}}$ of the function $f(x)=\|x\|$ where they exist.