In this paper, the original and general method for designing the first order digital FIR differentiators of even and odd length, with simultaneous approximation of the prescribed magnitude and group delay responses is presented. The proposed method represents an approach for FIR differentiator frequency response approximation, directly in the complex domain and is based on the least squares approximation method, with the originally modified eigenfilter method. It involves computing the elements and eigen-system of the quadratic, real and symmetric matrix, by the simultaneous minimization of the appropriate and originally defined quadratic measure error of the magnitude and group delay responses in the defined frequency bands. The given specifications of these two responses are incorporated in the minimization procedure. The eigenvector corresponding to the smallest eigenvalue from the computed matrix eigen-system presents the desired solution, i.e. the impulse response coefficients vector of the designed FIR differentiator. The weighting coefficients of the real and imaginary parts approximation, of the frequency response in the passband and stopband, are introduced. By the appropriate choice of these coefficient values, it is possible to affect the achieved approximation quality and accuracy. FIR differentiators, designed by this method do not posses neuther the antisymmetric feature of their impulse response coefficients, nor the strictly linear phase. Their passband group delay level is approximately constant and differs (lower or higher) from that of the corresponding linear phase FIR differentiators and can be varied in a relatively wide range. With the same length, differentiators designed by the proposed method have a lower passband magnitude response error than the corresponding mini-max differentiators. In order to illustrate its effectiveness, the numerical examples of their synthesis are also given.