Chapter 3: Differentiation Rules

Students learn multiple notations for derivatives (Leibniz, prime notation, and 𝑑/𝑑𝑥) and explore differentiability, noting the conditions where a function may fail to be differentiable due to sharp corners, cusps, vertical tangents, or discontinuities. Rules for differentiation are introduced systematically, including the Constant Rule, Power Rule, Constant Multiple Rule, and Sum and Difference Rules, allowing efficient calculation of derivatives for basic functions. The chapter also covers the Product Rule and Quotient Rule for combining functions, and extends differentiation to trigonometric functions such as sine, cosine, and tangent, with proofs grounded in limit properties. Higher-order derivatives are introduced, showing how repeated differentiation models acceleration and other changes in rates. Students also learn about the Chain Rule for differentiating composite functions, supported by practical examples from physics, biology, and economics. Implicit differentiation is presented as a powerful technique for functions defined implicitly rather than explicitly. The chapter concludes with applications of derivatives, including motion analysis (position, velocity, acceleration), slopes of curves, and rates of change in various sciences. By the end, readers have mastered both the conceptual meaning and the computational techniques of differentiation, preparing them for optimization, curve analysis, and further applications in calculus.