This paper establishes connections between two derivative estimation techniques: infinitesimal perturbation analysis (IPA) and the likelihood ratio or score function method. We introduce a systematic way of expanding the domain of the former to include that of the latter, and show that many likelihood ratio derivative estimators are IPA estimators obtained in a consistent manner through a special construction. Our extension of IPA is based on multiplicative smoothing. A function with discontinuities is multiplied by a smoothing complement, a continuous function that takes the value zero at a jump of the first function. The product of these functions is continuous and provides an indirect derivative estimator after an appropriate normalization. We show that, in substantial generality, the derivative of a smoothing complement is a randomized score function: its conditional expectation is a derivative of a likelihood ratio. If no conditional expectation is applied, derivative estimates based on multiplicative smoothing have higher variance than corresponding estimates based on likelihood ratios.