  We investigate the effect of non-canonical kinetic terms on inflation in
supergravity. We find that the biggest impact of such higher-derivative kinetic
terms is due to the corrections to the potential that they induce via their
effect on the auxiliary fields, which now have a cubic equation of motion. This
is in contrast to the usual (non-supersymmetric) effective field theory
expansion which assumes that mass-suppressed higher-derivative terms do not
affect the lower-derivative terms already present. We demonstrate with several
examples that such changes in the potential can significantly modify the
inflationary dynamics. Our results have immediate implications for effective
descriptions of inflation derived from string theory, where higher-derivative
kinetic terms are generally present. In addition we elucidate the structure of
the theory in the parameter range where there are three real solutions to the
auxiliary field's equation of motion, studying the resulting three branches of
the theory, and finding that one of them suffers from a singularity in the
speed of propagation of fluctuations.
