Gradient and Hessian of vector function

Question

I have the following function for an ATV problem,

$\psi(x)= \lVert y-x \rVert^{2} + \mu \sum_{i=1}^{n-1} \sqrt{\alpha^{2} + (x_{i+1} - x_{i})^{2}}$

(where $x,y \in \mathbb{R}^{n}$) I don’t know how to get the $\nabla \psi$ and Hessian. I’m very confused with the index terms.

Answer 1

I get the following gradient, but I'm not sure if it's correct

$$\frac{\partial \psi}{\partial x_{i}}=2(x_{i}-y_{i})-\mu\frac{x_{i+1}-x_{i}}{\sqrt{\alpha^{2}+(x_{i+1}-x_{i})^{2}}}$$

$$\frac{\partial \psi}{\partial x_{i+1}}=\mu\frac{x_{i+1}-x_{i}}{\sqrt{\alpha^{2}+(x_{i+1}-x_{i})^{2}}}$$

for each $1\leq i \leq n-1$