CalcRef

Derivatives · Basic Rules

Power Rule

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}

Bring the exponent down as a coefficient and reduce the exponent by one. Works for any real number n.

Conditions. n can be any real number. For n = 0 the result is 0. For integer n the rule holds wherever x^n is defined; for non-integer n, x^n and the rule require x > 0.
  • 3 worked examples
  • 24 in Derivatives

Worked examples

Find d/dx[x⁵].
  1. Apply power rule: 5x⁴

Answer: 5x⁴

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Find d/dx[x^(-2)].

Step-by-step solution locked.

Find d/dx[√x].

Step-by-step solution locked.

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Common questions

What is the Power Rule?

Bring the exponent down as a coefficient and reduce the exponent by one. Works for any real number n. It is one of the derivatives formulas in the CalcRef reference.

When does the Power Rule apply?

The Power Rule holds under this condition: n can be any real number. For n = 0 the result is 0. For integer n the rule holds wherever x^n is defined; for non-integer n, x^n and the rule require x > 0.

How do you use the Power Rule?

Worked example. Find d/dx[x⁵]. Apply power rule: 5x⁴. Answer: 5x⁴.

Practice

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