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AP Calc AB · Chapter 2 of 8

Differentiation: Definition and Basic Rules

Build the derivative from first principles, interpret derivative values, and use core differentiation rules for common function forms.

Why this chapter matters

The derivative is the main tool for describing instantaneous rate of change and local linear behavior.

What you will learn

  • Compute derivatives from the limit definition in simple cases.
  • Interpret derivative values and units in context.
  • Differentiate polynomial, power, exponential, and trigonometric functions using basic rules.

Lessons in this chapter

  1. Derivative as a limit of difference quotientsMove from average rate of change to instantaneous rate of change. Read the full guide →
  2. Notation and tangent line meaningUse f'(x), dy/dx, and derivative values to describe local behavior.
  3. Power, sum, and constant multiple rulesDifferentiate efficiently without rebuilding limits each time. Read the full guide →
  4. Derivatives of trig and exponential functionsApply standard derivatives and combine them in mixed expressions.

Study task

For s(t) = t^3 - 6t^2 + 9t, compute the average rate of change on [1, 3] and the instantaneous rate at t = 2, then compare meanings.

Chapter checkpoint

Using derivative rules, find d/dx of 3x^4 - 5x^2 + 7.

The derivative is 12x^3 - 10x.