  The natural forms of the Leibniz rule for the $k$th derivative of a product
and of Fa\`a di Bruno's formula for the $k$th derivative of a composition
involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$
rather than $d^k/dx^k$, with no assumptions about whether the variables
$x_1,...,x_k$ are all distinct, or all identical, or partitioned into several
distinguishable classes of indistinguishable variables. Coefficients appearing
in forms of these identities in which some variables are indistinguishable are
just multiplicities of indistinguishable terms (in particular, if all variables
are distinct then all coefficients are 1). The computation of the
multiplicities in this generalization of Fa\`a di Bruno's formula is a
combinatorial enumeration problem that, although completely elementary, seems
to have been neglected. We apply the results to cumulants of probability
distributions.
