Derivatives and Extrema: Theory and Applications

This lecture covers the theory of derivatives, including the calculation of derivatives for various functions such as trigonometric functions, polynomials, and composite functions. The instructor discusses the concept of local extrema and the conditions for a function to have a local maximum or minimum. The lecture also explores the application of derivatives in finding maximum and minimum points of functions on closed intervals, using the Mean Value Theorem and Rolle's Theorem. Additionally, the lecture delves into hyperbolic functions, their derivatives, and the concept of hyperbolic tangents and cotangents. The importance of finding critical points and verifying extremum points is emphasized through theoretical explanations and practical examples.