  We review the approach to calculation of one-loop effective action in ${\cal
N}=2,4$ SYM theories. We compute the non-holomorphic corrections to low-energy
effective action (higher derivative terms) in ${\cal N}=2$, SU(2) SYM theory
coupled to hypermultiplets on a non-abelian background for $R_{\xi}$-gauge
fixing conditions. A general procedure for calculating the gauge parameters
depending contributions to one-loop superfield effective action is developed.
The one-loop non-holomorphic effective potential is exactly found in terms of
the Euler dilogarithm function for a specific choice of gauge parameters.We
also discuss the calculations of hypermultiplet dependence of ${\cal N}=4$ SYM
effective action.
