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In this brief, modifications of the Schneider operator and the Al-Alaoui-Schneider-Kaneshige-Groutage rule have been explored for the improved performance of the fractional-order differentiator (FOD) in the low-frequency range. The FOD models are obtained using continued-fraction expansion (CFE), and it is observed that the magnitude responses obtained using the CFE outperform the results of the discretizations of FODs based on existing first-order and higher order s-to-z transformations in the low-frequency range. The phase responses of the FOD models show a linear response over a part of the low-frequency ranges that can be used for various applications. MATLAB simulation results have been presented to validate the effectiveness of the proposed work. These models can be used for hardware realizations of fractional-order systems.