Rates of Change: Derivatives
A derivative measures how fast a function's output changes with respect to its input — it is the instantaneous rate of change at any point. Written as f'(x) or dy/dx, the derivative is the slope of the tangent line to the curve at a given point. Derivatives are used to analyze motion, optimize systems, and model change in science, engineering, and economics.
Components of Derivatives
This section covers the fundamental derivative rules and techniques:
- Power Rule: For f(x) = xⁿ, the derivative is f'(x) = nxⁿ⁻¹.
- Sum & Constant Rules: The derivative of a sum is the sum of the derivatives; constants factor out, and the derivative of a constant alone is 0.
- Product & Quotient Rules: Product rule: (fg)' = f'g + fg'. Quotient rule: (f/g)' = (f'g - fg') / g².
- Chain Rule: For composite functions, d/dx[f(g(x))] = f'(g(x)) × g'(x) — differentiate the outer function and multiply by the derivative of the inner.
Examples of Derivatives
Power Rule Examples
- Find the derivative of f(x) = x⁴: f'(x) = 4x³.
- Find the derivative of f(x) = 3x⁵: f'(x) = 15x⁴.
- Find the derivative of f(x) = x² + 6x - 2: f'(x) = 2x + 6.
Sum & Constant Rules Examples
- Derivative of f(x) = 5x³ - 2x + 7: f'(x) = 15x² - 2 (the constant 7 drops out).
- Derivative of f(x) = 4x² + 3x: f'(x) = 8x + 3.
- Derivative of f(x) = 10: f'(x) = 0 (a constant has zero rate of change).
Product & Quotient Rule Examples
- Find d/dx of x² × sin(x): Using the product rule, 2x × sin(x) + x² × cos(x).
- Find d/dx of (3x + 1)(x² - 4): Product rule gives 3(x² - 4) + (3x + 1)(2x) = 3x² - 12 + 6x² + 2x = 9x² + 2x - 12.
- Find d/dx of x/(x + 1): Quotient rule gives ((1)(x + 1) - x(1)) / (x + 1)² = 1/(x + 1)².
Chain Rule Examples
- Find d/dx of (2x + 3)⁵: Outer derivative is 5(2x + 3)⁴, inner derivative is 2, so the result is 10(2x + 3)⁴.
- Find d/dx of √(x² + 1): Rewrite as (x² + 1)^(1/2), then 1/2 × (x² + 1)^(-1/2) × 2x = x/√(x² + 1).
- Find d/dx of sin(3x): The outer derivative gives cos(3x), multiplied by the inner derivative 3, giving 3cos(3x).