Using a directional form of constraint qualification weaker than Robinson’s, we derive an implicit function theorem for inclusions and use it for first- and second-order sensitivity analyses of the value function in perturbed constrained optimization. We obtain Holder and Lipschitz properties and, under a,to-gap condition, first-order expansions for exact and approximate solutions. As an application, differentiability properties of metric projections in Hilbert spaces are obtained, using a condition generalizing polyhedricity. We also present in the appendix a short proof of a generalization of the convex duality theorem in Banach spaces.