  We extend the theory of cosmological perturbations to the case when the
``matter'' Lagrangian is an arbitrary function of the scalar field and its
first derivatives. In particular, this extension provides a unified description
of known cases such as the usual scalar field and the hydrodynamical perfect
fluid. In addition, it applies to the recently proposed k-inflation, which is
driven by non-minimal kinetic terms in the Lagrangian. The spectrum of quantum
fluctuations for slow-roll and power law k-inflation is calculated. We find,
for instance, that the usual ``consistency relation'' between the tensor
spectral index and the relative amplitude of scalar and tensor perturbations is
modified. Thus, at least in principle, k-inflation is phenomenologically
distinguishable from standard inflation.
