  We introduce inflationary models where the inflaton features a field
dependent non-minimal derivative coupling to the Einstein tensor, that we name
GNMDC. This Horndeski term gives new and distinguishable inflationary
predictions in a framework that ameliorates possible problems related with
gradient instabilities during the reheating stage. We examine the inflationary
phenomenology using power law monomial and exponential potentials. We further
elaborate on the implications of the inflaton-modulated GNMDC and construct
GNMDC terms that can amplify the power spectrum of primordial perturbations at
small scales, triggering PBH production. An attractive feature of the GNMDC is
that inflation, as well as PBH production, can be implemented utilizing the
Higgs potential.
