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Explore the fundamental concepts of differentiation in this real analysis lecture that defines what it means for a function to be differentiable and establishes the essential connection between differentiability and continuity. Learn the basic algebraic laws governing differentiation, including Leibniz's rule for products, the quotient rule for ratios of functions, and the chain rule for composite functions. Master these foundational tools before advancing to more sophisticated results like Rolle's lemma and its application in proving the mean value theorem, a cornerstone result in calculus that describes the relationship between a function's values and its derivative over an interval. Gain rigorous mathematical understanding of these concepts through formal proofs and theoretical development that forms the backbone of advanced calculus and mathematical analysis.
