The author considers following problem from sensitivity analysis, i.e., how to calculate by means of Monte Carlo methods values of \(\partial /\partial x\int H(y)d\mu_ x(y)\) where \(\mu_ x\) is a family of probability measures depending on x and H(y) is a real valued function. This is a typical problem of stochastic optimization, when looking for \(\int H(y)d\mu_ x(y):=\min !)\) \((resp.:=\max !)\) etc. For this purpose a new procedure is presented based on the sampling of the weak derivative of a probability measure \(\mu_ x\) with the help of a pair of random variables representing the positive and negative part of this derivative, respectively. The proposed approach is illustrated by the transformation method, the rejection method and methods for generating discrete probabilities or discrete mixtures, respectively.