We consider a subclass of the Horndenski theory for studying cosmic inflation. In particular, we study a model in which the derivative self-interaction of the scalar field and the non-minimal derivative coupling to gravity sector are present in the Lagrangian. In order for the derivative self-interaction to be not too small and to have appreciable effects, we assume a special condition. We compute the observable quantities including the power spectra of scalar and tensor modes, the spectral indexes, and the tensor-to-scalar ratio using the aforementioned condition as well as the usual slow-roll approximations. As a result, we nd that the assumed relation reduces the tensor-to-scalar ratio r. In addition, we placed observational constraints on natural inflation and show that, for a certain range of the model parameter, the theoretical prediction of natural inflation isconsistent with the current observational data mainly due to the suppressed tensor-to-scalar ratio.