  We give two new proofs that the SRB measure of a C^2 path f_t of unimodal
piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the
topological class of f_0. The arguments are more conceptual than the one in our
previous paper, but require proving Holder continuity of the infinitesimal
conjugacy (a new result, of independent interest) and using spaces of bounded
p-variation. The first new proof gives differentiability of higher order if f_t
is smooth enough and stays in the topological class of f_0 and if the
observable smooth enough (a new result). In addition, this proof does not
require any information on the decomposition of the SRB measure into regular
and singular terms, making it potentially amenable to extensions to higher
dimensions. The second new proof allows us to recover the linear response
formula (i.e., the formula for the derivative at 0) obtained in our previous
paper and gives additional information on this formula.
