Linear response formula for piecewise expanding unimodal maps

The average R (t) of a smooth function with respect to the SRB measure of a smooth one-parameter family fₜ of piecewise expanding interval maps is not always Lipschitz. We prove that if fₜ is tangent to the topological class of f₀, then R (t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.