  Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian
manifold. In this paper, we prove differential Harnack inequalities for
positive solutions of nonlinear parabolic equations of the type $$\ppt f=\Delta
f-f \ln f +Rf.$$ We also comment on an earlier result of the first author on
positive solutions of the conjugate heat equation under the Ricci flow.
