  In this paper the Hartree equation is derived from the $N$-body
Schr\''odinger equation in the mean-field limit, with convergence rate
estimates that are uniform in the Planck constant $\hbar$. Specifically, we
consider the two following cases:(a) T\''oplitz initial data and Lipschitz
interaction forces, and (b) analytic initial data and interaction potential,
over short time intervals independent of $\hbar$. The convergence rates in
these two cases are $1/\sqrt{\log\log N}$ 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. It provides a derivation of the Vlasov
equation from the $N$-body classical dynamics using BBGKY hierarchies instead
of empirical measures.
