This paper discusses an accurate numerical approach of computing the Jacobian Matrix for the calculation of low dimensional manifolds for kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a highly stiff reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size, h.