Suppose $\displaystyle f: \mathbb{D} \rightarrow \mathbb{D}$ is an analytic function from the unit disk into itself with a fixed point at $\displaystyle z_0 \in \mathbb{D}$. Show that the stretching at $\displaystyle z_0$ of $\displaystyle f(z)$ in the hyperbolic metric is the same as the stretching at $\displaystyle z_0$ of $\displaystyle f(z)$ in the Euclidean metric,

$\displaystyle \displaystyle \lim_{ z \rightarrow z_0} \frac{\rho(f(z), z_0)}{\rho(z, z_0)} = \displaystyle \lim_{ z \rightarrow z_0} \frac{|f(z) - z_0|}{|z - z_0|} = |f'(z_0)|$.

There is a hint in the back of the book that says that $\displaystyle \rho(f(z), z_0) \approx \frac{2 |f(z)-z_0|}{1-|z_0|^2}$ and $\displaystyle \rho(z, z_0) \approx \frac{2 |z-z_0|}{1-|z_0|^2}$. However, I am still not sure how to prove this. I would appreciate a few hints. Thanks.