The Power Rule
“Power rule: bring it down, knock it down by one.”
The formula
d/dx xⁿ = n·xⁿ⁻¹How to read it: to differentiate xⁿ, bring the exponent n down front and drop the exponent by one
- n
- — the exponent of the power
- d/dx
- — the symbol for 'differentiate with respect to x'
- xⁿ⁻¹
- — the power with its exponent lowered by 1
The hook
No need to crank the limit definition every time — for powers it's one move: 'bring down, step down'.
In plain words
The slope function of xⁿ is: multiply out front by the exponent n, and lower the exponent to n−1.
The intuition
The bigger the exponent, the steeper the curve climbs. Differentiating pulls that 'climbing rate (n)' out front and drops the degree one notch to shape the slope.
How it's built
Two moves, that's all — ① bring the exponent down as a factor, and ② subtract 1 from the exponent. The rule just compresses the pattern you'd get by running the limit definition on x³, x⁴, and so on.
Example
Differentiate x⁵: exponent 5 comes down front, exponent becomes 4 → 5x⁴. At x=1 the slope is 5.
Common mistake
It's easy to forget to subtract 1 from the exponent. Also, the derivative of a constant is 0 — a constant is x⁰ with n=0, so the front factor is 0.
Where it's used
Differentiating any polynomial is basically this rule applied term by term. Since most models in physics and economics are polynomials, you use it daily.
Where it came from
Newton pulled out this pattern by expanding (1+h)ⁿ with the binomial theorem; later it was shown to hold for rational and real exponents too.
Prerequisites
Quick check
What is the derivative of x⁶?
- 6x⁵✓
- 6x⁷
- x⁵
- 5x⁶
Practice
What is the derivative of f(x)=x⁴ at x=2?
Answer: 32
- f'(x)=4x³
- Plug in x=2 → 4·8 = 32
Takeaway: Bring down, step down, then substitute once.
What is the derivative of f(x)=x²+x³ at x=1? (term by term)
Answer: 5
- f'(x)=2x+3x²
- Plug in x=1 → 2 + 3 = 5
Takeaway: The derivative of a sum is each term differentiated, added up.
Differentiate f(x)=5x² and plug in x=3.
Answer: 30
- The constant 5 rides along: f'(x)=5·2x=10x
- Plug in x=3 → 30
Takeaway: A constant coefficient rides straight through the derivative.