In this paper we develop an alternative method to derive finite difference approximations of derivatives. The purpose is to find schemes which work for a broader range of frequencies than the usual approximations based on polynomial fitting and Taylor´s Theorem to the expense of less accuracy for low frequencies. The numerical schemes are obtained as solutions to constrained optimizations problems in a weighted L2- norm in the frequency domain. We examine the accuracy of these schemes and compare them with the standard approximations. We also use the same approach to derive numerical schemes for time integration for differential equations with time independent operators. To test the accuracy of the different schemes, we study dispersion errors for a simple wave equation in one space dimension. We examine the number of points per wave length which is needed in order for the relative error in the phase velocity to be below a certain bound. A similar examination is carried out for the different time integration schemes.