  With a view to study the convergence properties of the derivative expansion
of the exact renormalization group (RG) equation, I explicitly study the
leading and next-to-leading orders of this expansion applied to the
Wilson-Polchinski equation in the case of the $N$-vector model with the
symmetry $\mathrm{O}(N) $. As a test, the critical exponents $% \eta $ and $\nu
$ as well as the subcritical exponent $\omega $ (and higher ones) are estimated
in three dimensions for values of $N$ ranging from 1 to 20. I compare the
results with the corresponding estimates obtained in preceding studies or
treatments of other $\mathrm{O}(N) $ exact RG equations at second order. The
possibility of varying $N$ allows to size up the derivative expansion method.
The values obtained from the resummation of high orders of perturbative field
theory are used as standards to illustrate the eventual convergence in each
case. A peculiar attention is drawn on the preservation (or not) of the
reparametrisation invariance.
