Derivatives in Calculus
What are Derivatives?
Derivatives represent the rate of change of a function with respect to a variable. They are fundamental to calculus and have applications in physics, engineering, economics, and many other fields.
Key Concepts:
- Definition of a derivative
- Differentiation rules
- Chain rule
- Product and quotient rules
- Applications of derivatives
1. Definition of a Derivative
The derivative of a function f(x) at a point x is defined as:
f'(x) = limh→0 [f(x + h) - f(x)] / h
Geometric Interpretation
The derivative represents the slope of the tangent line to the function at a given point.
2. Basic Differentiation Rules
Power Rule
If f(x) = xn, then f'(x) = nxn-1
Constant Rule
If f(x) = c, then f'(x) = 0
Sum Rule
[f(x) + g(x)]' = f'(x) + g'(x)
Product Rule
[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)
3. Common Derivatives
| Function | Derivative |
|---|---|
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| ex | ex |
| ln(x) | 1/x |
Practice Problems
Problem 1
Find the derivative of f(x) = 3x² + 2x - 5