  In this work we aim to revive the interest for non-minimal derivative
coupling theories of gravity, in light of the GW170817 event. These theories
include a string motivated non-minimal kinetic term for the scalar field of the
form $\sim \xi(\phi) G^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$ and predict
that the primordial tensor perturbations have a speed that it is distinct from
the speed of light. Due to the fact that the Universe is classical during and
in the post-inflationary epoch, there is no fundamental reason for the graviton
to change its mass, the GW170817 event severely constrained these theories. We
analyze and formalize the inflationary phenomenology of these theories, using
both the latest Planck data and the GW170817 event data to constrain these
theories. Due to the fact that there is no constraint in the choice of the
scalar potential and the non-minimal coupling function, we provide several
classes of viable models, using convenient forms of the minimal coupling
function in terms of the scalar potential, aiming for analyticity, and we
discuss the advantages and disadvantages of each viable model.
