Calculus

Study of continuous change through derivatives and integrals.

Limits and ContinuityDerivatives and Rates of ChangeIntegrals and Area Under Curves
Applications of DerivativesTechniques of IntegrationInfinite Series and Convergence
Modeling MotionEconomics and OptimizationBiology and Population Growth
Breaking Down ProblemsMemorizing Key FormulasChecking Your Work
Limits and Continuit...Derivatives and Rate...Integrals and Area U...Applications of Deri...Techniques of Integr...Infinite Series and ...Modeling MotionEconomics and Optimi...Biology and Populati...Breaking Down Proble...Memorizing Key Formu...Checking Your Work
Basic Concepts

Derivatives and Rates of Change

What Is a Derivative?

Derivatives measure how quickly something is changing. In math terms, the derivative of a function at a point tells you the slope of the tangent line at that point. It's like asking, "How fast is this car going, exactly right now?"

The Notation

Derivatives are written as \( f'(x) \) or \( \frac{df}{dx} \). This reads as "the derivative of \( f \) with respect to \( x \)."

The Power Rule

For powers of \( x \), the derivative is simple: \[ \frac{d}{dx}x^n = n x^{n-1} \]

Why Use Derivatives?

  • To find where a function increases or decreases
  • To locate maximum or minimum points (great for optimization!)
  • To analyze motion, growth, and change in science and nature

Key Formula

\[\frac{d}{dx} x^n = n x^{n-1}\]

Examples

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

  • The speedometer in a car shows the derivative of your position over time (your velocity).

In a Nutshell

Derivatives tell us how fast things are changing at any instant.

Key Terms

Derivative
The instantaneous rate of change of a function.
Slope
The steepness of a line; in calculus, the derivative at a point.
Next
Integrals and Area Under Curves