Differentiability: Foundations of Calculus

Understanding differentiability is the cornerstone of advanced calculus, providing the essential tools for optimization, analysis of rates of change, and modeling complex systems. This text-only course transforms your conceptual understanding of functions, allowing you to confidently analyze their behavior, identify critical points, and solve challenging mathematical problems involving derivatives. What you'll learn: * Understand the rigorous definition of differentiability using limits and its geometric interpretation. * Master the crucial relationship between continuity and differentiability in various types of functions. * Apply fundamental differentiation rules, including the Chain Rule, Product Rule, and Quotient Rule. * Practice analyzing complex functions, such as piecewise definitions and those involving absolute values, for points of non-differentiability. * Learn advanced techniques for solving problems involving implicit differentiation and parametric forms. The course begins by establishing a strong theoretical foundation, reviewing necessary concepts from limits and continuity. We then systematically explore the formal definition of the derivative, followed by detailed written explanations and worked examples of rules and problem-solving strategies. This course is designed for beginners who have a basic grasp of pre-calculus and algebra, and who are ready to tackle foundational college-level calculus concepts. No prior advanced calculus knowledge is required. Start building your strong mathematical foundation today.