  It is well known that the Leibniz rule for the integer derivative of order
one does not hold for the fractional derivative case when the fractional order
lies between 0 and 1. Thus it poses a great difficulty in the calculation of
fractional derivative of given functions as well as in the analysis of
fractional order systems. In this work, we develop a few fractional
differential inequalities which involve the Caputo fractional derivative of the
product of continuously differentiable functions. We establish some of their
properties and propose a few propositions. We show that these inequalities play
a very essential role in the Lyapunov stability analysis of nonautonomous
fractional order systems.
