We consider a subclass of Horndeski theories for studying cosmic inflation. In particular, we investigate models of inflation in which the derivative self-interaction of the scalar field and the non-minimal derivative coupling to gravity are present in the action and equally important during inflation. In order to control contributions of each term as well as to approach the single-term limit, we introduce a special relation between the derivative interaction and the coupling to gravity. By calculating observable quantities including the power spectra and spectral tilts of scalar and tensor perturbation modes, and the tensor-to-scalar ratio, we found that the tensor-to-scalar ratio is suppressed by a factor of $(1+1/gamma)$, where $gamma$ reflects the strength of the derivative self-interaction of the inflaton field with respect to the derivative coupling gravity. We placed observational constraints on the chaotic and natural inflation models and showed that the models are consistent with the current observational data mainly due to the suppressed tensor-to-scalar ratio.
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