Considering a second-order patch surrounding an axial reference object point, formulas for the second eld derivatives of the wavefront aberration function, corresponding to rays both in the tangential and sagittal sections, are given. To obtain these derivatives, Coddingtons equations are used in a way alternative to that employed to calculate the second aperture derivatives. The wavefront aberration function for any point in the patch is written in terms of data acquired tracing tangential rays from the axial point alone. The eå ectiveness of the procedures is tested numerically in two photographic objectives. The plots for the eld derivatives can be incorporated to the traditional ones to improve the global optimization of the optical system.