For functions, $f_1, f_2, ...$, which are all convex, the sub-differential of their sum is the sum of their sub-differentials,
$\partial \sum\limits_{i} f_i = \sum_\limits{i} \partial f_i$
What could be said about the sub-differential if one or more of these functions are non-convex? Let us ignore the question of whether it is possible to obtain the sub-differential of a particular non-convex function or not.