This paper provides strong bounds on perturbations over a collection of independent random variables, where 'strong' has to be understood as uniform w.r.t. some functional norm. Our analysis is based on studying the concept of weak differentiability. By applying a fundamental result from the theory of Banach spaces, we show that weak differentiability implies norm Lipschitz continuity. This result leads to bounds on the sensitivity of finite products of probability measures, in norm sense. We apply our results to derive bounds on perturbations for the transient waiting times in a G/G/1 queue.