  We consider interacting $N$-Bosons in three dimensions. It is known that the
difference between the many-body Schr\"odinger evolution in the mean-field
regime and the corresponding Hartree dynamics is of order $1/N$. We investigate
the time dependence of the difference. To have sub-exponential bound, we use
the results of time decay estimate for small initial data. We also refine time
dependent bound for singular potential using Strichartz estimate. We consider
the interaction potential $V(x)$ of type $\lambda\exp(-\mu|x|)|x|^{-\gamma}$
for $\lambda\in\mathbb{R}$, $\mu\geq0$, and $0<\gamma<3/2$, which covers the
Coulomb and Yukawa interaction.
