  We study fermions derivatively coupled to axion-like or pseudoscalar fields,
and show that the axial vector current of the fermions is not conserved in the
limit where the fermion is massless. This apparent violation of the classical
chiral symmetry is due to the background axion field. We compute the
contributions to this anomalous Ward identity due to the pseudoscalar field
alone, which arise in Minkowski space, as well as the effects due to an
interaction with an external gravitational field. For the case of massless
fermions, these interactions induce terms in the axion effective action that
can be removed by the addition of local counterterms. We demonstrate that these
counterterms are generated by the transformation of the path integral measure
when transforming the theory from a form where the chiral symmetry is manifest
to one where the symmetry is only apparent after using the classical equations
of motion. We work perturbatively in Minkowski space and include the effects of
interactions with a linearized gravitational field. Using the heat kernel
method, we study the transformation properties of the path integral measure,
and include the effects of non-linear gravity as well as interactions with
gauge fields. Finally, we verify our relation by considering derivatively
coupled fermions during pseudoscalar-driven inflation and computing the
divergence of the axial current in de Sitter spacetime.
