 Partial subdifferential

For a convex scalar function $f(x,y)$, the partial subdifferential ${\partial }_{y}f(x,y)$ means the subdifferential of $v\mapsto f(x,v)$ at $v=y$. It is not differentiation of a set-valued subdifferential. If this section is differentiable, the partial subdifferential is the singleton containing ${\nabla }_{y}f(x,y)$.