About Course

Unlock the power of calculus by diving deep into the very heart of instantaneous change. In this focused mini-course, you’ll journey from the intuitive notion of “rate of change” all the way to the formal limit definition of the derivative, then conquer every fundamental shortcut—constant, power, sum/difference, and constant-multiple rules. You’ll finish by exploring higher-order derivatives for polynomials, gaining insights into acceleration and concavity. Packed with real-world analogies, step-by-step derivations, and targeted practice, this module lays the bedrock for AP Calculus AB success and every application you’ll face beyond.

What Will You Learn?

  • Grasp instantaneous rate of change through real-world analogies
  • Derive the limit definition of the derivative from first principles
  • Simplify difference quotients using algebraic techniques
  • Apply constant, power, sum/difference, and constant-multiple rules with confidence
  • Compute and interpret higher-order derivatives for polynomial functions

Course Content

1. Conceptual Foundations of the Derivative
Begin with vivid real-life scenarios—like a car’s speedometer needle or the changing slope of a ski trail—to internalize what “instantaneous rate of change” really means. You’ll see how secant lines morph into tangent lines and how average rates hone in on exact values as intervals shrink.

  • 1.1 Secant Line & Average Rate of Change
  • 1.2 Tangent Line & Instantaneous Rate of Change
  • 1.3 Notation & Interpretation: f′(x), dy/dx, Leibniz vs. Lagrange

2. Formal Limit Definition of the Derivative
Translate intuition into precision by mastering the difference quotient. We’ll derive the limit definition from first principles, then practice algebraic simplifications—cancelling h’s and rationalizing expressions—to compute derivatives by hand

3. Fundamental Differentiation Rules
Speed up your workflow with a complete toolbox of rules. Learn why a constant’s derivative is zero, how linearity lets you pull constants out, and why the power rule reigns supreme for any polynomial. You’ll practice each rule until it’s second nature.

4. Higher-Order Derivatives of Polynomials
Push beyond the first derivative to uncover acceleration, concavity, and the elegant patterns behind successive derivatives. You’ll compute second, third, and nth derivatives of polynomials, then connect these to graphical behavior and motion models.

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