  We consider slow-roll inflationary models in a class of modified theories of
gravity which contains non-minimal curvature-inflaton couplings, i.e., the
$f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the
inflaton energy-momentum tensor. On top of the minimally coupled $T$ that has
been widely investigated in the literature, we further include a $RT$ mixing
term in the theory. This mixing term introduces non-minimal derivative
couplings and plays an important role in inflationary dynamics. Taking chaotic
and natural inflation as examples, we find that the predictions for spectral
tilt and the tensor-to-scalar ratio are sensitive to the existence of the $RT$
mixing term. In particular, by turning on this mixing term, it is possible to
bring chaotic and natural inflation into better agreement with observational
data.
