Leibniz rule for the high q -derivatives of a quotient of two functions

In this paper, we introduce some explicit formulas for the positive integer order q -derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between the commutative algebras and the properties of q -operators. Especially, the third formula is a beautiful one involving the determinant of the functions and their q -derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, the last formula is used for proving of some interesting identities. MSC CLASSIFICATION: 05A30, 26A24