Divergence operator on the hyperbolic plane $\mathbb{H}^2$

Question

I want to figure out the difference between the divergence operator on $\mathbb{R}^2$ which I denote by $\text{div}_{\mathbb{R}^2}$ and the divergence operator $\text{div}_{\mathbb{H}^2}$ on the hyperbolic plane in two dimensions \begin{align} \mathbb{H}^2=\{(x,y)\in \mathbb{R}_+\times \mathbb{R}\}. \end{align} So what I am interested if possible is to get a representation of $\text{div}_{\mathbb{H}^2}$ in terms of $\partial_x$ and $\partial_y$, i.e. \begin{align} \text{div}_{\mathbb{H}^2}=\text{div}_{\mathbb{R}^2}+...? \end{align} Unfortunately I'm really far from being an expert in differential geometry. If this is possible it would be really kind to exemplary demonstrate how one can derive it if possible :)