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Power Rule Derivative

Also known as derivative of x^n · d/dx x^n · power rule · differentiate a power · slope of a power function

ddx xn=n xn−1\frac{d}{dx}\,x^{n} = n\,x^{n-1}

Units aren’t used in this calculation — every value is a plain number.

Worked example: slope of x³ at x = 4 → 48 — press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

See also
Related Rates: Inflating Sphere — The power rule doing a job: dV/dr for a sphere is 4πr², and the chain rule turns it into a flow rate.Power of a Power — The exponent law underneath xⁿ⁻¹, if the algebra rather than the calculus is the sticking point.

Learning zone

The power rule is the first derivative anyone learns and the one every other differentiation rule leans on: bring the exponent down in front, then knock one off it. Worked example: the slope of y = x³ at x = 4 is 3 × 4² = 3 × 16 = 48. Check it against the definition from either side — between x = 3.999 and x = 4.001 the curve rises from 63.952 to 64.048, a gain of 0.096 over 0.002, which is 48.0 to three figures. That agreement is the whole content of the rule: the messy limit of (x+h)³ − x³ over h collapses, once you expand the bracket and cancel, to 3x².

It holds for every real exponent, not just whole ones, and that generality is Newton's. His generalised binomial theorem of 1665 let him expand (1+x)n(1+x)^{n} for fractional and negative n, which turned √x into x^0.5 and 1/x² into x⁻² and put them under the same rule — so the derivative of √x is ½x^(−1/2) = 1/(2√x), and the derivative of 1/x is −x⁻² = −1/x². Pierre de Fermat had been computing these slopes by his method of adequality thirty years earlier without the notation to state a general rule. Two traps. The rule is about x raised to a constant, never a constant raised to x: the derivative of 2ˣ is 2ˣ ln 2, a different animal entirely. And the exponent n = 0 is not a special case needing memorisation — x⁰ is the constant 1, and the rule's own factor of 0 out front correctly reports a slope of zero everywhere.

Power Rule Derivative
ddx xn=n xn−1\frac{d}{dx}\,x^{n} = n\,x^{n-1}
Where
  • dydx\frac{dy}{dx}= Derivative at x
  • nn= Exponent
  • xx= Value of x
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