When re-scaling velocities with a rescaling_factor ($A$), distances $r$ in the velocity expression should be re-scaled as well:
$\tilde r=r A$
$\tilde v_r = v_r A$
${\rm d}\tilde v_r / {\rm d}\tilde r = A^{-1} \ {\rm d} (v_rA) / {\rm d}r = {\rm d} v_r / {\rm d}r$,
where $A$ is either a function of AP parameters or a free parameter, and re-scaled quantities are marked with tilde.