What's a Derivative?

Derivatives tell us how functions change. Think of them as the ultimate way to describe speed, growth, or decline—the rate at which something is happening at any given moment.

The Concept

The derivative of a function at a point measures how fast the function's value is changing as its input changes. The process of finding a derivative is called differentiation.

How We Write It

The derivative of \( f(x) \) is often written as \( f'(x) \) or \( \frac{df}{dx} \).

Why Are Derivatives Useful?

• They help us find the slope of tangent lines to curves.
• They allow us to calculate velocity, acceleration, and other rates of change.
• They're essential in optimization problems (like maximizing profits or minimizing costs).

How to Find Derivatives

• Use the definition: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
• Apply basic rules: power, product, quotient, and chain rules.