Explore calculus through the lens of integral polynumbers in this comprehensive video lecture. Begin with a review of Pascal's Array and binomial coefficients before delving into the Taylor bipolynumber of polynumbers. Learn how to derive the standard form for polynomial derivatives and discover a lesser-known formula for second subderivatives. Master important rules for finding subderivatives of sums and products of polynumbers. Gain insights into alternative approaches to calculus that challenge traditional mathematical education methods. Study notation for binomial coefficients and examine proofs for key theorems. This fundamental lecture offers a fresh perspective on calculus, suitable for both teachers and students seeking a deeper understanding of mathematical foundations.