Local neocortical circuits are characterized by stereotypical physiological and structural features that subserve generic computational operations. These basic computations of the cortical microcircuit emerge through the interplay of neuronal connectivity, cellular intrinsic properties, and synaptic plasticity dynamics. How these interacting mechanisms generate specific computational operations in the cortical circuit remains largely unknown. Here, we identify the neurophysiological basis of both the rate of change and anticipation computations on synaptic inputs in a cortical circuit. Through biophysically realistic computer simulations and neuronal recordings, we show that the rate-of-change computation is operated robustly in cortical networks through the combination of two ubiquitous brain mechanisms: short-term synaptic depression and spike-frequency adaptation. We then show how this rate-of-change circuit can be embedded in a convergently connected network to anticipate temporally incoming synaptic inputs, in quantitative agreement with experimental findings on anticipatory responses to moving stimuli in the primary visual cortex. Given the robustness of the mechanism and the widespread nature of the physiological machinery involved, we suggest that rate-of-change computation and temporal anticipation are principal, hard-wired functions of neural information processing in the cortical microcircuit.