Find the subdifferential of the following function $$f(w)=\frac{1}{2}\left \| w \right \|^2_2+C\cdot \sum_{i=1}^{m} \max(0,1-\alpha _i \cdot x_i^Tw)$$ where $C>0$, $w, x_i \in \Bbb R^n$ and $\alpha_i \in \Bbb R$.
I've found the first part $$\partial f(w) = \frac{1}{2}\cdot2w + \cdots$$ but have no idea what to do with the second one.