  In this article we develop a simplistic approach to revisit the classical
Kato-Ponce inequality, which is also known as 'fractional Leibniz rule.' As a
consequence, we derive the validity of this inequality even in quasi-Banach
spaces $L^p$ for $p<1$ with a certain restrictions on the indices. Also, we
display the sharpness of this restriction by means of a counter-example.
Finally, we also prove a multi-parameter variant of the inequality, which
allows for partial fractional derivatives on $R^n$.
