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A reconfigurable photonic signal processing system for arbitrary-order temporal differentiation of broadband microwave waveforms, with bandwidths up to a few tens of gigahertz, is proposed and experimentally demonstrated. This technique enables full programmability of the differentiation operator to be applied on the input microwave signal, including any desired linear differential-equation operator, by suitably reshaping the incoherent power spectrum according to the corresponding finite-difference time-domain (FDTD) equations obtained from the Euler's approximation. Successive photonic time derivatives of Gaussian-like pulse intensity waveforms with pulse widths of 42 and 72 ps were accurately achieved up to the second and the fourth order, respectively, using the same photonic processing platform. A more general operator, conceived to directly emulate the second-order differential-equation modeling a high-frequency series resistor, inductor, and capacitor (RLC) circuit, was also implemented and successfully tested.