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

\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 \rho(f(z), z_0) \approx \frac{2 |f(z)-z_0|}{1-|z_0|^2} and \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.