We propose a differential phase contrast imaging method in x-ray microscopy by utilizing a biased derivative filter, which is structurally similar to that used in visible optics, except that phase changes by the filter cannot be ignored in the x-ray range. However, it is demonstrated that the filter's phase retardation does not disturb its function of phase contrast imaging, and even enhances the signals to some extent. Theoretical formulations and corresponding numerical simulations show that the approach is capable of performing characteristic differential microscopic phase imaging with nanometer-scale resolution. Manageable parameters are also examined in detail for pursuing a high image quality.