Calculus
Study of continuous change through derivatives and integrals.
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.