The rules of differentiation
What you'll learn
Differentiate any polynomial fast with the power, constant, and sum rules — no limits required.
Computing every derivative from the limit definition would be exhausting. Thankfully, a handful of rules let you differentiate most functions by inspection — and they all come from that limit, proved once so you don't have to repeat it.
The core rules
| Rule | If f(x) = | then f′(x) = |
|---|---|---|
| Power | xⁿ | n·xⁿ⁻¹ |
| Constant | c | 0 |
| Constant multiple | c·g(x) | c·g′(x) |
| Sum | g(x) + h(x) | g′(x) + h′(x) |
The power rule is the workhorse: bring the exponent down in front, then subtract one from it. The constant rule makes sense visually — a constant is a flat line, slope 0. The sum rule means you can differentiate term by term.
Worked example
Differentiate f(x) = 4x³ − 2x² + 7x − 5, one term at a time:
- 4x³ → 4·3x² = 12x²
- −2x² → −2·2x = −4x
- 7x → 7·1 = 7 (since x = x¹, the exponent drops to x⁰ = 1)
- −5 → 0 (constant)
So f′(x) = 12x² − 4x + 7. A cubic differentiates to a quadratic; each term loses one degree.
The power rule isn't just for whole numbers
The exponent n can be negative or fractional, which quietly covers roots and reciprocals:
- √x = x^(1/2) → (1/2)x^(−1/2) = 1 / (2√x)
- 1/x = x^(−1) → −1·x^(−2) = −1/x²
Rewrite the expression as a power of x first, then the power rule just works.
Why this matters
These rules turn differentiation from a limit computation into algebra you can do in your head. They're the vocabulary for everything ahead — the moment a problem says "take the derivative," this is what you actually reach for.
Check your understanding
Question 1 of 2
Using the power rule, d/dx of x⁵ is: