Anti-Derivatives and a Generalized Primitive Formula for Function Products

This publication introduces a novel method for determining the primitive (anti-derivative) of any function products using a deterministic and algebraic formula. By defining new operations on both derivatives and anti-derivatives, the approach simplifies and systematizes the computation of primitives. As a key result, we derive an explicit formula for the n-th anti-derivative of a product of functions, denoted (f · g)(−n) , analogous to the classical Leibniz rule for derivatives (f · g)(n) . This new formula parallels the structure of the Binomial Series Expansion for negative exponents, (a + b)−n , just as the Leibniz rule aligns with Newton’s binomial theorem for numbers. The proposed framework provides a unified perspective connecting differentiation, integration, and binomial identities, and marks a significant advancement in the symbolic treatmentof anti-derivatives.