The power rule

The shortcut

Last lesson, finding a derivative meant working out a limit by hand. For powers of x there's a shortcut that skips all of that. The power rule says:

d/dx xⁿ = n·xⁿ⁻¹

In words: the power comes down in front as a multiplier, and the new power is one less than the old one.

Watch it work

| Function | Bring the power down | Subtract one | Derivative | |---|---|---|---| | x³ | 3x³ | 3x² | 3x² | | x² | 2x² | 2x¹ | 2x | | x¹⁰ | 10x¹⁰ | 10x⁹ | 10x⁹ |

When the new power is 1, you just write x, since x¹ = x. That's why x² becomes 2x.

Check it against the limit

Last lesson, the limit gave the slope of y = x² at x = 3:

lim (h→0) [(3 + h)² − 9] / h = lim (h→0) (6 + h) = 6

With the power rule, the derivative of x² is 2x, so at x = 3 the slope is 2 · 3 = 6. Same answer, way faster. The rule doesn't replace the limit definition; it's proved from it, once, so you don't have to redo the limit every time.

A number in front rides along

If there's a number multiplying the power, keep it and multiply. This is the constant multiple rule:

d/dx 5x⁴ = 5 · 4x³ = 20x³

Worked example: find the slope of y = 2x³ at x = 1. The derivative is 2 · 3x² = 6x². At x = 1, that's 6 · 1² = 6.

Try it yourself

1. Find the derivative of x⁵. Then find its value at x = 1.
2. Find the derivative of 3x². What's the slope at x = 2?

(Answers: 1. 5x⁴, which is 5 at x = 1. 2. 6x, so the slope at x = 2 is 12.)

Good to know

• The power rule d/dx xⁿ = n·xⁿ⁻¹ holds for every real exponent n (for the x where xⁿ is defined); this lesson uses whole-number powers only.
• The constant multiple rule (a number in front stays as a factor) is what "rides along" means.
• "With no limits at all" means no limit work by hand; the rule itself is proved from the limit definition of the derivative.
• Checks: x³ → 3x², x² → 2x, x¹⁰ → 10x⁹, 5x⁴ → 20x³, and 2x at x = 3 is 6, matching the limit from last lesson.

Answer to the comment question

f(x) = x⁴. What's f′(2)? f′(2) = 32. By the power rule, f′(x) = 4x³. At x = 2: 4 · 2³ = 4 · 8 = 32.