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In this paper, an efficient design of FIR digital differentiator using the $L_1$-optimality criterion is proposed. We present a technique based on the modified Newton method to solve the design problem so that the optimal differentiator coefficients are obtained by minimizing the absolute error. The novel $L_1$-error function leads to a flat response at low-frequencies. Extensive simulations are carried out to validate the proposed design. The superiority of the proposed design is evident by comparing it with other conventional design techniques such as, windowing, minimax and the least-squares approach.