The paper presents the general approximation strategy for derivatives of solutions obtained by the parametric integral equation system (PIES). The proposed approach is applied to calculating derivatives in linear and nonlinear torsion of a bar. The nonlinear problem is solved iteratively and requires repeated calculating of integrals in order to determine stresses or strains. Due to elimination of the repeated integration, the proposed strategy increases the efficiency of calculations. Furthermore, it ensures high accuracy of solutions to the considered problem, which has been tested in comparison to the analytical and/or numerical solutions reported in the literature obtained by other methods.