Implicit Differentiation: Solving Equations Locally

This lecture discusses implicit differentiation, focusing on solving equations locally when x and y are linked by a constraint. Through examples like f(x, y) = x³ + 12x²y² - 3sin(y) = 8, the instructor demonstrates how to find functions of y and x. The Implicit Function Theorem is introduced, showing how to find unique functions g(x) defined on a neighborhood of (a, b) when D₂f(a, b) ≠ 0. The lecture emphasizes the concept of local invertibility and the Jacobian determinant in determining the uniqueness of functions.