AP Calculus AB

Advanced Placement Calculus AB covering limits, derivatives, and integrals.

Understanding LimitsIntroduction to DerivativesFundamentals of Integrals
Techniques of DifferentiationApplications of DerivativesDefinite Integrals and the Fundamental Theorem
Calculus in Physics and EngineeringCalculus in Economics and BiologyCalculus and Everyday Problems
Mastering Multiple-Choice QuestionsShowing Work and Earning Free Response Points
Understanding LimitsIntroduction to Deri...Fundamentals of Inte...Techniques of Differ...Applications of Deri...Definite Integrals a...Calculus in Physics ...Calculus in Economic...Calculus and Everyda...Mastering Multiple-C...Showing Work and Ear...
Basic Concepts

Introduction to Derivatives

What Are Derivatives?

Derivatives measure how fast something changes—think of them as the mathematical version of speed! If you want to know how your position changes over time, the derivative gives you the velocity.

The Formal Definition

The derivative of a function \( f(x) \) at a point \( x \) is defined as:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

How Do We Find Derivatives?

  • Power Rule: \( \frac{d}{dx} x^n = nx^{n-1} \)
  • Product Rule: \( (fg)' = f'g + fg' \)
  • Quotient Rule: \( \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \)
  • Chain Rule: Used for composite functions.

What Do Derivatives Tell Us?

  • Slope of the tangent line at a point
  • Instantaneous rate of change

Why Should You Care?

Derivatives help us model motion, growth, and lots of real-world phenomena!

Examples

  • If \( f(x) = x^2 \), then \( f'(x) = 2x \).

  • The slope of the curve \( y = \sin(x) \) at any point is \( \cos(x) \).

In a Nutshell

Derivatives show how a function is changing at any point.

Key Terms

Derivative
A measure of how a function's output changes as its input changes.
Tangent Line
A straight line that touches a curve at a single point and has the same slope as the curve at that point.
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Fundamentals of Integrals