Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely many C² functions, and the subgradients of the indicator of a set defined by finitely many C² constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.