As an enhanced version of existing results on Kac's propagation of chaos, which describes the convergence of mean-field particle systems to a system of independent McKean-Vlasov particles as the number of particles tends to infinity, we prove the convergence at the level of derivatives with respect to the perturbations of the initial values and the driving noises, together with explicit convergence rates that can be sharp. As a consequence, the intrinsic derivative with respect to the initial distribution of a single particle converges to that of an independent McKean-Vlasov SDE.