Inverse Rule for Formal Power Series

just a couple of thoughts. the approach you indicate seems more useful and elegant than one based on, say $$ \frac1{1-xP} = 1 +xP+xP^2+\cdots $$ another method might be to use $$ \begin{align} D^1(f^{-1}) &= -f^{-2}f_1 \\ D^2(f^{-1}) &= -f^{-3}(ff_2-2f_1^2) \\ &\cdots \end{align} $$ to build a McLaurin expansion

As closed a form as I could get I posted here. It looks pretty ugly... but I'm not sure how much prettier it can get.