  It is well-known that the coefficients in Faa di Bruno's chain rule for
higher derivatives can be expressed via numeration of partitions. It turns out
that this has a natural form as a formula for the vector case. To this formula
two proofs are presented, both "explaining" its form involving partitions: one
as a purely algebraic fact, and one "from first principles" for the case of
Frechet derivatives of mappings between Banach spaces.
