We propose and demonstrate a tunable fractional-order photonic differentiator (DIFF) that can process input pulses with a sub-gigahertz bandwidth. Our scheme utilizes the self-induced optical modulation effect observed in a silicon-on-insulator micro-ring resonator. Gaussian-like pulses with varying pulse widths between 7.5 and 20 ns are employed for differentiation, achieving an energy efficiency over 45%, to the best of our knowledge, which surpasses all previously reported schemes for input pulses with a sub-gigahertz bandwidth. We simulate the temporal dynamics of pulses to gain insight into the physical mechanisms underlying the differentiated outputs and provide a method for differentiation order adjustment, which is experimentally realized using an all-optical pump-probe technique.