  We present a direct calculation for the first derivative of the isovector
nucleon form factors with respect to the momentum transfer $q^2$ using the
lower moments of the nucleon 3-point function in the coordinate space. Our
numerical simulations are performed using the $N_f = 2 + 1$ nonperturbatively
$O(a)$-improved Wilson quark action and Iwasaki gauge action near the physical
point, corresponding to the pion mass $M_\pi =138$ MeV, on a (5.5 fm)$^4$
lattice at a single lattice spacing of $a = 0.085$ fm. In the momentum
derivative approach, we can directly evaluate the mean square radii for the
electric, magnetic, and axial-vector form factors, and also the magnetic moment
without the $q^2$ extrapolation to the zero momentum point. These results are
compared with the ones determined by the standard method, where the $q^2$
extrapolations of the corresponding form factors are carried out by fitting
models. We find that the new results from the momentum derivative method are
obtained with a larger statistical error than the standard method, but with a
smaller systematic error associated with the data analysis. Within the total
error range of the statistical and systematic errors combined, the two results
are in good agreement. On the other hand, two variations of the momentum
derivative of the induced pseudoscalar form factor at the zero momentum point
show some discrepancy. It seems to be caused by a finite volume effect, since a
similar trend is not observed on a large volume, but seen on a small volume in
our pilot calculations at a heavier pion mass of $M_{\pi}= 510$ MeV.
Furthermore, we discuss an equivalence between the momentum derivative method
and the similar approach with the point splitting vector current.
