  In probabilistic coherence spaces, a denotational model of probabilistic
functional languages, morphisms are analytic and therefore smooth. We explore
two related applications of the corresponding derivatives. First we show how
derivatives allow to compute the expectation of execution time in the weak head
reduction of probabilistic PCF (pPCF). Next we apply a general notion of
"local" differential of morphisms to the proof of a Lipschitz property of these
morphisms allowing in turn to relate the observational distance on pPCF terms
to a distance the model is naturally equipped with. This suggests that
extending probabilistic programming languages with derivatives, in the spirit
of the differential lambda-calculus, could be quite meaningful.
