  An assumption of smooth response to small parameter changes, of statistics or
long-time averages of a chaotic system, is generally made in the field of
sensitivity analysis, and the parametric derivatives of statistical quantities
are critically used in science and engineering. In this paper, we propose a
numerical procedure to assess the differentiability of statistics with respect
to parameters in chaotic systems. We numerically show that the existence of the
derivative depends on the Lebesgue-integrability of a certain density gradient
function, which we define as the derivative of logarithmic SRB density along
the unstable manifold. We develop a recursive formula for the density gradient
that can be efficiently computed along trajectories, and demonstrate its use in
determining the differentiability of statistics. Our numerical procedure is
illustrated on low-dimensional chaotic systems whose statistics exhibit both
smooth and rough regions in parameter space.
