  Escort distributions have been shown to be very useful in a great variety of
fields ranging from information theory, nonextensive statistical mechanics till
coding theory, chaos and multifractals. In this work we give the notion and the
properties of a novel type of escort density, the differential-escort
densities, which have various advantages with respect to the standard ones. We
highlight the behavior of the differential Shannon, R\'enyi and Tsallis
entropies of these distributions. Then, we illustrate their utility to prove
the monotonicity property of the LMC-R\'enyi complexity measure and to study
the behavior of general distributions in the two extreme cases of minimal and
very high LMC-R\'enyi complexity. Finally, this transformation allows us to
obtain the Tsallis q-exponential densities as the differential-escort
transformation of the exponential density.
