Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the deriative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.