  Leibniz's rule for the $n$-th derivative of a product is a very well known
and extremely useful formula. Unfortunately, currently, there does not exist a
similar explicit formula for the $n$-th derivative of a quotient. In this
article, we introduce an 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.
