Derivative of a coordinate of a vector w.r.t. its norm

Question

Let $x \in \mathbb R^n$ and $\|x\| = \sqrt{x_1^2 + \dots + x_n^2}$ the usual euclidean norm. Does it make sense to consider the derivative of a coordinate w.r.t. the norm, i.e. can I define $$ \dfrac{\partial x_1}{\partial \|x\|} $$ and how should I calculate? Also if I fix another point $y \in \mathbb R^n$, does it makes sense to calculate $$\dfrac{\partial \| x- y\|}{\partial \|x\|} \quad ?$$

Answer 1

Hint:

In spherical coordinates:

\begin{equation} \begin{array}{c} \| \mathbf{x}\| = r \\ x_1 = r \text{cos}(\phi_1) \\ x_2 = r \text{sin}(\phi_1)\text{cos}(\phi_2) \\ x_2 = r \text{sin}(\phi_1)\text{sin}(\phi_2)\text{cos}(\phi_3) \\ \vdots \\ x_{n-1} = r \text{sin}(\phi_1)\cdots \text{sin}(\phi_{n-2})\text{cos}(\phi_{n-1}) \\ x_{n} = r \text{sin}(\phi_1)\cdots \text{sin}(\phi_{n-2})\text{sin}(\phi_{n-1}), \\ \end{array} \end{equation} where $\phi_1, \phi_2, \ldots, \phi_n$ are the angular coordinates and $r$ the radius of the hypersphere.