Explore an algebraic infinitesimal approach to product and chain rules in calculus without relying on limits or real numbers. Learn about dual complex numbers and bi-polynumbers as foundational concepts for understanding derivatives in a general field F. Discover the Derivative theorem and its applications to the Product Rule and Chain Rule in a broad mathematical context. Gain insights into this novel approach to calculus, which will be further developed in the Algebraic Calculus Two online course.