The availability of high precision observational data in cosmology means that it is possible to go beyond simple descriptions of cosmic inflation in which the expansion is driven by a single scalar field. One set of models of particular interest involve the Dirac-Born-Infeld (DBI) action, arising in string cosmology, in which the dynamics of the field are affected by a speed limit in a manner akin to special relativity. In this talk, I will introduce a scalar-tensor theory in which the matter component is a field with a DBI action. Transforming to the Einstein frame, I will explore the effect of the resulting coupling on the background dynamics of the fields and the first-order perturbations. The coupling forces the scalar field into the minimum of its effective potential, so the dynamics are determined by the DBI field, which has the interesting effect of increasing the number of efolds of inflation and decreasing the boost factor of the DBI field. Focusing on this case, I will show that the power spectrum of the primordial perturbations is determined by the behaviour of the perturbations of the modified DBI field and calculate the effect of varying the model parameters on the inflationary observables.