Power, Constant Multiple, Sum, and Difference Rules
Prove and apply the Power Rule, constant multiple rule, and sum/difference rules. Differentiate any polynomial and rational function (rewritten with negative exponents) in one pass.
A lesson of AP Calculus AB, in Differentiation: Definition and Fundamental Properties, about 10 minutes.
Basic Differentiation Rules
By the end of this lesson, you will be able to apply the power rule, constant rule, sum and difference rules, and constant multiple rule to find derivatives efficiently.
The Core Rules of Differentiation
Once you understand what a derivative means (the instantaneous rate of change), you need efficient rules to calculate them without computing limits every time. These rules cover the vast majority of functions you'll encounter in AP Calculus AB.
The Power Rule
For any function f(x) = x^n, where n is any real number:
$$f'(x) = nx^{n-1}$$
Process: Multiply by the exponent, then reduce the exponent by 1.
Examples:
| Function | Derivative | Process |
|---|---|---|
| f(x) = x⁵ | f'(x) = 5x⁴ | Multiply by 5, reduce exponent to 4 |
| f(x) = x³ | f'(x) = 3x² | Multiply by 3, reduce to 2 |
| f(x) = x | f'(x) = 1 | x = x¹; 1·x⁰ = 1 |
| f(x) = x⁻² | f'(x) = -2x⁻³ | Works for negative exponents |
| f(x) = x^(1/2) | f'(x) = ½x^(-1/2) | Works for fractional exponents |
Important: The power rule applies to any real exponent — positive, negative, or fractional.
The Constant Rule
The derivative of a constant is zero:
$$\frac{d}{dx}[c] = 0$$
Intuition: A constant function has no rate of change — its graph is a horizontal line with slope 0.
Examples:
- f(x) = 7 → f'(x) = 0
- f(x) = -π → f'(x) = 0
The Constant Multiple Rule
$$\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$$
Constants factor out of derivatives.
Examples:
- f(x) = 5x³ → f'(x) = 5 · 3x² = 15x²
- f(x) = -3x⁴ → f'(x) = -3 · 4x³ = -12x³
- f(x) = ½x⁶ → f'(x) = ½ · 6x⁵ = 3x⁵
The Sum and Difference Rules
$$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$$