Partial Derivatives: Composition of Functions

This lecture covers the concept of partial derivatives in the context of composing functions. It explains how to calculate the derivative of a composition of functions and proves the continuity of the resulting function. Through demonstrations and theorems, the instructor illustrates the process of computing limits and applying the mean value theorem. The lecture also discusses the existence of increments using the mean value theorem and extends the theory to cases where certain assumptions are relaxed. Notations commonly used in the literature are presented, emphasizing the relationship between independent and dependent variables. Additionally, the derivative of an integral with parameters varying in the integrand and limits is explored, showcasing the application of partial derivatives in integral calculus.