This note studies the relationship between two important approaches in perturbation analysis (PA)-perturbation realization (PR) and weak derivatives (WDs). Specifically, we study the relation between PR and WDs for estimating the gradient of stationary performance measures of a finite state-space Markov chain. Will show that the WDs expression for the gradient of a stationary performance measure can be interpreted as the expected PR factor where the expectation is carried out with respect to a distribution that is given through the weak derivative of the transition kernel of the Markov chain. Moreover, we present unbiased gradient estimators.