Explicit Formula for the n-th Derivative of a Quotient

TL;DR

This paper derives an explicit formula for the n-th derivative of a quotient of two functions, extending Leibniz's rule, and applies it to obtain new partition identities and expressions for special derivatives.

Contribution

It introduces a novel explicit formula for the n-th derivative of a quotient, expanding the toolkit for differential calculus and combinatorial identities.

Findings

Derived an explicit formula for the n-th derivative of a quotient.

Applied the formula to obtain new partition identities.

Developed expressions for certain special derivatives.

Abstract

Leibniz's rule for the $n$-th derivative of a product is a very well known and extremely useful formula. In this article, we introduce an analogous explicit formula for the $n$-th derivative of a quotient of two functions. Later, we use this formula to derive new partition identities and to develop expressions for some special $n$-th derivatives.