In this paper, the design of a digital differentiator is investigated. First, the relation between fractional sample delay filter and digital differentiator is established such that the differentiator can be obtained from the fractional delay filter by using the computation of limit. Then, conventional finite impulse response (FIR), allpass and Farrow-based fractional delay filters are directly applied to design the digital differentiator. Finally, several design examples are illustrated to demonstrate the effectiveness of this new design approach.