Analytic subordination results are powerful technical tools in free harmonic analysis. As Voiculescu observed, analytic subordination for free additive convolution is intimately related to the coalgebra of the free difference quotient, which is a comultiplication that is also a derivation. Here we will consider two other derivation-comultiplications appearing in free probability theory, and discuss their relation to analytic subordination for free multiplicative convolution and for freely Markovian triples.