Abstract
In this paper, we introduce some explicit and recursive formulas for the th ‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of ‐differential operator. One of the formulas involves the determinant of the functions and their ‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting ‐identities.
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