The Chain Rule and Chaos

This video is a sequel to my "Measurement and Calculus: Continuity and Derivatives through the Lens of Interval Arithmetic" video    • Measurement and Calculus: Continuity and D...   . It covers how the chain rule is a natural consequence of interpreting the derivative as an uncertainty multiplication factor, and what that tells us about real-world systems where the same function gets applied repeatedly. Wikipedia has an excellent article on Chaos Theory: https://en.wikipedia.org/wiki/Chaos_t..., discussing the additional properties a system needs besides sensitivity to be considered chaotic. For instance, it needs a property called "topological mixing", which formalizes the idea that if we start out at any value with any uncertainty to it, and repeatedly apply our function, we eventually could wind up at pretty much any value. Riffle shuffle demonstration video is public domain: https://commons.wikimedia.org/wiki/Fi... You can read more about card shuffling models in Mann's "How Many Times Should You Shuffle a Deck of Cards?" https://chance.dartmouth.edu/teaching... I've built some visualizers for my model of shuffling a deck of cards: https://trkern.github.io/shuffle