  In this paper we study the influence of including snapshots that approach the
velocity time derivative in the numerical approximation of the incompressible
Navier-Stokes equations by means of proper orthogonal decomposition (POD)
methods. Our set of snapshots includes the velocity approximation at the
initial time from a full order mixed finite element method (FOM) together with
approximations to the time derivative at different times. The approximation at
the initial velocity can be replaced by the mean value of the velocities at the
different times so that implementing the method to the fluctuations, as done
mostly in practice, only approximations to the time derivatives are included in
the set of snapshots. For the POD method we study the differences between
projecting onto $L^2$ and $H^1$. In both cases pointwise in time error bounds
can be proved. Including grad-div stabilization both in the FOM and POD methods
error bounds with constants independent on inverse powers of the viscosity can
be obtained.
