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With a view to study the convergence properties of the derivative expansionof the exact renormalization group (RG) equation, I explicitly study theleading and next-to-leading orders of this expansion applied to theWilson-Polchinski equation in the case of the $N$-vector model with thesymmetry $\mathrm{O}(N) $. As a test, the critical exponents $% \eta $ and $\nu$ as well as the subcritical exponent $\omega $ (and higher ones) are estimatedin three dimensions for values of $N$ ranging from 1 to 20. I compare theresults with the corresponding estimates obtained in preceding studies ortreatments of other $\mathrm{O}(N) $ exact RG equations at second order. Thepossibility of varying $N$ allows to size up the derivative expansion method.The values obtained from the resummation of high orders of perturbative fieldtheory are used as standards to illustrate the eventual convergence in eachcase. A peculiar attention is drawn on the preservation (or not) of thereparametrisation invariance.Comment: Dedicated to Lothar Sch\"afer on the occasion of his 60th birthday.  Final versio

Wilson–Polchinski exact renormalization group equation for O (N) systems: leading and next-to-leading orders in the derivative expansion