The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and Poisson processes to derive these representations in a unified way and to provide a probabilistic interpretation for the derivatives. By similar variational methods, we obtain apparently new integro-differential identities which the density of a strictly $α$-stable multivariate density satisfies. Then, we extend Crofton's derivative formula known in integral geometry to the case of a Poisson process. Finally we use this extension to give a new probabilistic proof of a version of this formula for binomial point processes.