The Hadamard semidifferential retains the advantages of the differential calculus such as the chain rule and semiconvex functions are Hadamard semidifferentiable. The semidifferential calculus extends to subsets of $${\mathbb {R}}^n$$ without Euclidean smooth structure. This set-up is an ideal tool to study the semidifferentiability of objective functions with respect to families of sets which are non-linear non-convex complete metric spaces. Shape derivatives are differentials for spaces endowed with Courant metrics. Topological derivatives are shown to be semidifferentials on the group of Lebesgue measurable characteristic functions.