What is the derivative of a constant?

Zero. A constant does not change, so its rate of change is zero.

Does the power rule work for negative or fractional exponents?

Yes. The same rule applies, so the derivative of x⁻¹ is −x⁻² and of x^(1/2) is ½x^(−1/2).

Grade 12 · Mathematics

Power & product rules

Why it matters

Differentiation rules.

Advanced20m readingPrerequisite: Definition

What is the power rule for derivatives?

Short answer

The power rule is a shortcut for differentiating powers of x: the derivative of xⁿ is n·xⁿ⁻¹ — multiply by the exponent and reduce the exponent by one. For example, the derivative of x⁵ is 5x⁴. Combined with the constant multiple and sum rules, it differentiates any polynomial quickly without using the limit definition each time.

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Frequently asked questions

What is the derivative of a constant?: Zero. A constant does not change, so its rate of change is zero.
Does the power rule work for negative or fractional exponents?: Yes. The same rule applies, so the derivative of x⁻¹ is −x⁻² and of x^(1/2) is ½x^(−1/2).

What is the power rule for derivatives?

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Frequently asked questions