A Full Course on Differentiation (Multivariable Calculus)

Differentiation in multivariable calculus is a fundamental tool for understanding how functions change in multiple dimensions, and this video provides a comprehensive, step-by-step guide to mastering it. We begin by introducing functions of multiple variables, setting the stage for analyzing surfaces, level curves, and how variables interact in higher dimensions. From there, we dive into partial derivatives, exploring how functions change with respect to individual variables while holding others constant. We introduce the chain rule and product rule, showing how they extend naturally to multiple variables and work through applied examples to reinforce their usage. We also derive the function differential using Taylor series, providing deeper insight into approximating functions locally and understanding rates of change in multidimensional space. As we progress, we tackle more advanced topics such as implicit functions, change of variables, and Jacobians—essential tools for transformations in multivariable calculus. We explain how Jacobians relate to inverse functions and why they play a crucial role in coordinate transformations. Optimization is a key application of multivariable differentiation, so we cover stationary points, global and local maxima and minima, and introduce the powerful Lagrange multipliers method for constrained optimization. Finally, we conclude with a discussion on exact differentials, tying everything together and demonstrating their significance in calculus and physics. By the end of this course, you'll have a solid grasp of multivariable differentiation, equipping you with the skills to tackle real-world problems in engineering, physics, and applied mathematics. Whether you're a student or a professional looking for a refresher, this video provides clear explanations, worked examples, and in-depth insights to help you master differentiation in multiple dimensions. 00:00:00 – Introduction - Functions of Multiple Variables 00:05:53 – Level Curves 00:10:55 – Derivatives of Multivariable Functions 00:19:19 – The Chain Rule 00:23:02 – The Product Rule 00:24:44 – Chain & Product Rules Applied to Multiple Variables 00:32:30 – Derivation of the Function Differential Using Taylor Series 00:45:46 – Rates of Change 00:55:58 – Implicit Functions 01:04:46 – Change of Variables 01:18:22 – Jacobians and Inverse Functions 01:50:36 – Stationary Points (Global & Local Maxima and Minima) 02:24:50 – Lagrange Multipliers 02:56:18 – Exact Differentials