Harmonic maps between Riemann surfaces

Question

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the equation \begin{equation} u_{z\overline{z}}+\frac{2\rho_u}{\rho}u_zu_{\overline{z}}=0 \end{equation} where the indices denote the derivatives. My question now is: what exactly is the $\rho_u$ looking like? Let's take $S_2$ as the hyperbolic plane equipped with the usual hyperbolic metric. Am I right in the assumption that $\rho_u$ is just $\frac{\partial 1/y}{\partial u}$ if we let $u=x+iy$? (The explicit formula for hyperbolic space is also one of the exercises in the book)