  We consider the solution to a stochastic differential equation with a drift
function which depends smoothly on some real parameter $\lambda$, and admitting
a unique invariant measure for any value of $\lambda$ around $\lambda$ = 0. Our
aim is to compute the derivative with respect to $\lambda$ of averages with
respect to the invariant measure, at $\lambda$ = 0. We analyze a numerical
method which consists in simulating the process at $\lambda$ = 0 together with
its derivative with respect to $\lambda$ on long time horizon. We give
sufficient conditions implying uniform-in-time square integrability of this
derivative. This allows in particular to compute efficiently the derivative
with respect to $\lambda$ of the mean of an observable through Monte Carlo
simulations.
