Calculus 12 · Mathematics · Lesson 3.2
Power & product rules
- Level
- Grade 12
- Reading time
- 14 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Differentiation rules.
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.
Concept preview
What you will be able to do
- 01Apply the power rule to differentiate powers of x, including negative and fractional exponents.
- 02Apply the constant-multiple, sum and difference rules to differentiate any polynomial.
- 03Apply the product rule to differentiate a product of two functions.
- 04Apply the quotient rule to differentiate a ratio of two functions.
- 05Derive the power rule from first principles and diagnose a flawed application of the rules.
Computing every derivative from the limit definition is correct but slow, and for a function built from several terms it quickly becomes unwieldy. Fortunately the first-principles limits fall into a small number of patterns, and once each pattern is proved once and for all it becomes a rule you may simply apply. This lesson collects the workhorse rules of differentiation: the power rule for a single power of the variable, the constant-multiple, sum and difference rules for combining terms, and the product and quotient rules for functions multiplied or divided together. Together they let you differentiate any polynomial, any rational function, and any algebraic combination of powers almost by inspection, reserving the limit definition for the rare cases the rules do not cover or for proving the rules themselves.
School Plan
Power & product rules
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Related topics
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).