Winding number: times f(w) goes around unit disc I have a problem which is like this:

Let $S^1=\partial D(0,1)$ and consider a differentiable function $f:S^1\rightarrow S^1$. Can you compute the number of times that $f(w)$ goes around $S^1$ per each time that $w$ goes around $S^1$?

I have tried to reparameterize the first $S^1$ with a $\gamma$ path and then parametrize the second $S^1$ in terms of $f(\gamma)$, but I do not get any further.
Any hints or ideas?

In fact, holomorphic functions/residue theory isn't required. See Baby Rudin, exercises 23-26 of ch. 8. A brief summary:
(1) Given a closed differentiable curve $\gamma:[a,b]\longrightarrow\Bbb C\setminus\{0\}$, let be
$$
\text{Ind}(\gamma) =
\frac1{2\pi i}\int_a^b\frac{\gamma'(t)}{\gamma(t)}\,dt.
$$
(yes, is a disguised line integral)
(2) Considering $\gamma\exp(-\int(\gamma'/\gamma))$ and the periodicity of the complex exponential is easy to prove that $\text{Ind}(\gamma)\in\Bbb Z$.
(3) Compute $\text{Ind}(\gamma)$ for the curves $\gamma(t) = \exp(int)$, $t\in[0,2\pi]$.
(4) Prove (essentially) that the integer-valued $\text{Ind}(\gamma)$ depends continuously of $\gamma$.
(5) Using (4) you can define $\text{Ind}(\gamma)$ for continuous curves.
