  We study the problem of conditional expectations in free random variables and
provide closed formulas for the conditional expectation of resolvents of
arbitrary non-commutative polynomials in free random variables onto the
subalgebra of an arbitray subset of the variables. More precisely, given a
linearization of the resolvent we compute a linearization of its conditional
expectation. The coefficients of the expressions obtained in this process
involve certain Boolean cumulant functionals which can be computed by solving a
system of equations. On the way towards the main result we introduce a
non-commutative differential calculus which allows to evaluate conditional
expectations and the said Boolean cumulant functionals. We conclude the paper
with several known and new examples which illustrate the working of the
developed machinery and two appendices. The first appendix contains a purely
algebraic approach to Boolean cumulants and the second appendix provides a
crash course on linearizations of rational series.
