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This manuscript presents a design of a differentiator in the digital domain with its low noise realization. It manifests the minimization of the L1 -error objective function by using a hybrid optimization technique consisting of the particle swarm and simulated annealing optimization algorithm. The obtained magnitude response provides a noteworthy approximation of the ideal differentiator with a minimal magnitude inaccuracy when compared with the existing designs. The realization structures are also investigated and compared in terms of the noise gain behavior.