  Within the framework of mappings between affine spaces, the notion of $n$-th
polarization of a function will lead to an intrinsic characterization of
polynomial functions. We prove that the characteristic features of derivations,
such as linearity, iterability, Leibniz and chain rules, are shared -- at the
finite level -- by the polarization operators. We give these results by means
of explicit general formulae, which are valid at any order $n$, and are based
on combinatorial identities. The infinitesimal limits of the $n$-th
polarizations of a function will yield its $n$-th derivatives (without
resorting to the usual recursive definition), and the above mentioned
properties will be recovered directly in the limit. Polynomial functions will
allow us to produce a coordinate free version of Taylor's formula.
