  We explore the possibility of a blue-tilted gravitational wave spectrum from
potential-driven slow-roll inflation in the Horndeski theory. In Kamada et al.
(2012), it was claimed that a blue gravitational wave spectrum cannot be
obtained from stable potential-driven slow-roll inflation within the Horndeski
framework. However, it has been demonstrated that the spectrum of primordial
gravitational waves can be blue in inflation with the Gauss-Bonnet term, where
the potential term is dominant and slow-roll conditions as well as the
stability conditions are satisfied. To fill in this gap, we clarify where the
discrepancy is coming from. We extend the formulation of Kamada et al. (2012)
and show that a blue gravitational wave spectrum can certainly be generated
from stable slow-roll inflation if some of the conditions previously imposed on
the form of the free functions in the Lagrangian are relaxed.
