  We use the differential algebra of simple polytopes to explain the remarkable
relation of the combinatorics of the associahedra and permutohedra with the
compositional and multiplicative inversion of formal power series. This
approach allows to single out the associahedra and permutohedra among all
graph-associahedra and emphasizes the significance of the differential
equations for polytopes derived earlier by one of the authors. We discuss also
the link with the geometry of Deligne-Mumford moduli spaces $\bar M_{0,n}$ and
provide a new interpretation of the combinatorics of cyclohedra in relation
with the classical Fa\`a di Bruno's formula.
