  The classic Cayley identity states that \det(\partial) (\det X)^s =
s(s+1)...(s+n-1) (\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of
indeterminates and \partial=(\partial/\partial x_{ij}) is the corresponding
matrix of partial derivatives. In this paper we present straightforward
combinatorial proofs of a variety of Cayley-type identities, both old and new.
The most powerful of these proofs employ Grassmann algebra (= exterior algebra)
and Grassmann-Berezin integration. Among the new identities proven here are a
pair of "diagonal-parametrized" Cayley identities, a pair of
"Laplacian-parametrized" Cayley identities, and the "product-parametrized" and
"border-parametrized" rectangular Cayley identities.
