In this paper, the closed-form design of fractional order differentiator (FOD) using discrete cosine transform (DCT) is presented. First, the definition of fractional derivative is reviewed briefly. Then, the DCT-based interpolation method is applied to compute the fractional derivative of a given digital signal. Next, the transfer function of FOD is obtained from the DCT computational results by using suitable index mapping. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design method.