  Gravitational particle production in the early universe is due to the
coupling of matter fields to curvature. This coupling may include derivative
terms that modify the kinetic term. The most general first order action
contains derivative couplings to the curvature scalar and to the traceless
Ricci tensor, which can be dominant in the case of (pseudo-)Nambu-Goldstone
bosons or disformal scalars, such as branons. In the presence of these
derivative couplings, the density of produced particles for the adiabatic
regime in the de Sitter phase (which mimics inflation) is constant in time and
decays with the inverse effective mass (which in turn depends on the coupling
to the curvature scalar). In the reheating phase following inflation, the
presence of derivative couplings to the background curvature modifies in a
nontrivial way the gravitational production even in the perturbative regime. We
also show that the two couplings -- to the curvature scalar and to the
traceless Ricci tensor -- are drastically different, specially for large
masses. In this regime, the production becomes highly sensitive to the former
coupling while it becomes independent of the latter.
