  Let the function $\varphi$ be holomorphic in the unit disk $\mathbb{D}$ of
the complex plane $\mathbb{C}$ and let $\varphi (\mathbb{D})\subset
\mathbb{D}$. We study the level sets and the critical points of the hyperbolic
derivative of $\varphi$,
$$|D_{\varphi}(z)|:=\frac{(1-|z|^2)|\varphi'(z)|}{1-|\varphi(z)|^2}.$$ In
particular, we show how the Schwarzian derivative of $\varphi$ reveals the
nature of the critical points.
