In this paper the Hartree equation is derived from the N-body Schrödinger equation uniformly in the Planck constant in two different cases, specifically (a) for Töplitz initial data and Lipschitz interaction force, and (b) for analytic initial data and interaction potential, and over short time intervals, independent of the Planck constant. The convergence rates in these two cases are 1/ log log N and 1/N respectively. The treatment of the second case is entirely self-contained and all the constants appearing in the final estimate are explicit. Moreover it provides a derivation of the Vlasov equation out of the N-body classical dynamics using hierarchies and not empirical measures.