I am looking at the gradient descent method for group lasso questions. Here's what I am currently stuck at.

Given the quadratic form of the objective function:
$$
f(x) = \frac{1}{2} x^T V x - m^T x + \sum_{l=1}^g \|x_{G_l}\|_2
$$
where $x$ and $m$ are vectors and $V$ is the positive semidefinite matrix. The $x_{G_l}$ is just referring to the vector of the elements in group $G_l$. The objective function is assumed to be differentiable and convex.

I was able to find the partial derivative for the first two components of the objective function:
$$
f'(x) = Vx - m
$$

However, I became confused with doing partial differentiation for group lasso. I know if I have $\|x\|_2^2$, I can easily differentiate it to $2x$.

Can anyone give an insight on how I should proceed?