Unit 3 Summary & Derivative Rules

AP Calculus · Axiom Academy

SUMMARY Unit 3: Differentiation Summary Master the foundation of calculus Unit 3 focused on differentiation - the process of finding rates of change. You learned both the theoretical foundation (limits) and practical shortcuts (rules) for computing derivatives. Limit Definition: - The precise mathematical definition Power Rule: - Works for any power of x Product Rule: - For multiplying functions Quotient Rule: - For dividing functions Chain Rule: - For composite functions (most important!) Trigonometric Derivatives: Sine → Cosine, Cosine → -Sine, Tan → Sec² Implicit Differentiation: Finding dy/dx when y isn't isolated Higher-Order Derivatives: f''(x), f'''(x), etc. - measuring rates of change of rates Differentiability vs Continuity: A function must be continuous to be differentiable, but continuity doesn't guarantee differentiability (corners, cusps) Order Matters: In quotient rule: "lo d-hi minus hi d-lo, square the bottom" Simplify First: When possible, algebraically simplify before differentiating Physics: Velocity (rate of change of position), Acceleration (rate of change of velocity) Biology: Population growth rates, disease spread modeling Economics: Marginal cost, marginal revenue, elasticity of demand Engineering: Optimization, motion control, heat dissipation You're now ready for applications of derivatives : Finding critical points and extrema (maxima and minima) Curve sketching with first and second derivatives

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