We consider the sensitivity, with respect to a parameter q, of parametric families of operators  vectors  corresponding to the adjoints  of  via  and one parameter semigroups  We display formulas relating weak differentiability of    to weak differentiability of  and  We give two applications: The first one concerns the sensitivity of the Ornstein-Uhlenbeck process with respect to its location parameter. The second one provides new insights regarding the Wright-Fisher diffusion for small mutation parameter.