Basic Concepts
In a nutshell: Derivatives measure how things change and help solve real-world problems.
## Rates of Change
Differentiation is the process of finding the derivative, which represents the rate of change of a function.
- The derivative of \( f(x) \), written \( f'(x) \), tells how \( f(x) \) changes as \( x \) changes.
- The power rule, product rule, quotient rule, and chain rule are essential techniques for finding derivatives.
## Applications
- Finding the slope of a tangent line at a point
- Analyzing motion (velocity and acceleration)
- Solving optimization problems
## Real-World Relevance
Derivatives are everywhere—speedometers, economics, biology, and more!
Examples
- The derivative of \( f(x) = x^2 \) is \( 2x \).
- If a car's position is \( s(t) \), its velocity is \( s'(t) \).