  In light of the cosmological observations, we investigate dark energy models
from the Horndeski theory of gravity. In particular, we consider cosmological
models with the derivative self-interaction of the scalar field and the
derivative coupling between the scalar field and gravity. We choose the
self-interaction term to have an exponential function of the scalar field with
both positive and negative exponents. For the function that has a positive
exponent, our result shows that the derivative self-interaction term plays an
important role in the late-time universe. On the other hand, to reproduce the
right cosmic history, the derivative coupling between the scalar field and
gravity must dominate during the radiation-dominated phase. However, the
importance of such a coupling in the present universe found to be negligible
due to its drastic decrease over time. Moreover, the propagation speed of
gravitational waves estimated for our model is within the observational bounds,
and our model satisfies the observational constraints on the dark energy
equation of state.
