Explore dual complex numbers and Leibniz's differentiation rules in this 27-minute mathematics lecture. Delve into a purely algebraic approach to infinitesimals that extends differential calculus to general fields, including finite fields. Learn about polynumbers, a data structure approach to polynomials, and how they can be evaluated using dual numbers. Examine Leibniz's rules of differentiation and their historical origins with Johann Faulhaber. Gain insights into the algebraic foundations of calculus and the innovative ways of understanding infinitesimals in mathematics.