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. (read more)