Showing the winding number of the unit circle is $1$

1. Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$.

2. I'm trying to compute $n(\gamma, 0)$ as follows:

$$n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 \pi i} \int_0^{2 \pi} {1 \over \gamma(t)} \gamma'(t)\ dt= \underbrace{\ldots}_{\text{?}} = 1$$

3. As flagged above, I'm not sure what justifies the inference to the answer of $1$.

Because $\gamma(t)=e^{it},\;t\in[0,2\pi]$. Hence $$\frac{1}{2\pi i}\int_0^{2\pi}\frac{\gamma'(t)}{\gamma(t)}\,dt =\frac{1}{2\pi i}\int_0^{2\pi}\frac{ie^{it}}{e^{it}}\,dt=1$$