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A derivative represents the instantaneous rate of change of a function with respect to a variable. In practical terms, it describes the slope of a tangent line at any given point on a curve. Core Differentiation Methods Chain Rule (Composite Functions): Used for nested functions, where the derivative is the "outside" derivative multiplied by the "inside" derivative.Implicit Differentiation: Applied when variables are intermingled (e.g., y^2 - 3xy + x^2 = 7), requiring both sides to be differenti

The study of derivatives is a cornerstone of calculus, marking the transition from static algebra to the mathematics of change. This chapter focuses on how functions behave at a specific moment, providing the tools to measure sensitivity, steepness, and motion. 1. The Concept of Instantaneous ChangeAt its core, a derivative represents the instantaneous rate of change of a function with respect to a variable. While algebra allows us to calculate the average speed over an interval, the derivative