We can define the degree of a function $f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$ as $$\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial \theta} d\theta$$ What is the meaning of $f^{-1}$ and $\frac{\partial f}{\partial \theta}$ ?

I read somewhere that $f^{-1} = \bar{f} \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$ and $\frac{\partial f}{\partial \theta} \in H^{-\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$. Why can we say that ?