Chapter 7: Differentiation

The core principle involves finding the gradient of a curve at any specific point through a rigorous limiting process where the gradient is defined as the limit of secant line gradients as the distance between endpoints approaches zero. This foundational approach, known as differentiation from first principles, establishes why the derivative represents the precise slope of a tangent line at a point rather than an approximation. Students learn multiple notational systems for expressing derivatives, including Leibniz notation which emphasizes the rate of change between variables and Lagrange notation which highlights the function-based perspective. The power rule forms the cornerstone of practical differentiation, allowing efficient computation of derivatives for polynomial and power functions by multiplying by the exponent and reducing it by one. Beyond individual functions, scalar multiple and additive rules extend this technique to linear combinations of functions, while the chain rule enables differentiation of composite functions by connecting rates of change through intermediate variables. The chapter demonstrates how derivatives directly apply to finding equations of tangent lines, which share the curve's slope at a point of contact, and normal lines that are perpendicular to tangents with slopes equal to negative reciprocals. The second derivative extends understanding further by measuring how the gradient itself changes across the domain, providing insight into the concavity and curvature of functions. These interconnected concepts create a comprehensive framework for analyzing function behavior and form the foundation for optimization and curve sketching in subsequent mathematics.