The Chain Rule: Intuition + Proof!

The chain rule is a fundamental principle in differential calculus to calculate the derivatives of composite functions. The principle is also the basis for u-substitution in integration and manifests as a "cancellation of differentials" in the equation dz/dy * dy/dx = dz/dx In this video, we prove the chain rule rigorously from first principles. In the process, we intuitively explain what differentiability of a function really means and what it really means for the tangent line to be a good linear approximation to the graph of a function. We also give a simple wrong "proof" of the chain rule and some interesting counterexamples. The video is beneficial to anyone learning calculus in high school and college (e.g., AP Calculus in the USA) as well as to anyone in the process of/interested in learning real analysis and proof-based math. If you enjoy the content on my channel, I would love if you could please like ✅ , subscribe 🎉, and share ⏩ with students, classmates, friends and family - it means so much to help support me in my goal of creating elite infinite free accessible math education worldwide! 😊 If you would like to support my channel and my long-term vision of changing lives through math education beyond this (which of course would mean so much to me 🤗), please check out my "Buy me a Coffee" and my "Patreon" page! (If I have provided value to you through my free videos, any level of contribution is appreciated!) Buy me a Hot Chocolate 😊: https://buymeacoffee.com/math_mastery... ☕️ Become a Patreon Member 🤩:   / mathmasterywithamitesh