In this paper we discuss the problem of constructing accurate numerical schemes for calculating state sensitivities for application to design. In computing sensitivities, spatial derivatives of finite element state approximations are required. We introduce two projection techniques, motivated by a-posteriori error estimators used in adaptive finite element schemes, for computing improved derivative approximations. We briefly describe the techniques and their implementation for calculating sensitivities for a 1-D model problem. Numerical results illustrating the improved accuracy of the sensitivities are presented for the 1-D model problem as well as for two dimensional flow around a cylinder.