Power rule for differentiation: d/dx (xⁿ) = nxⁿ⁻¹
The power rule for differentiation is d/dx (xⁿ) = nxⁿ⁻¹, and d/dx (axⁿ) = anxⁿ⁻¹.
What each letter means
- \(n\) the power: any number, including fractions and negatives
- \(a\) a constant multiple
- \(\frac{d}{dx}\) differentiate with respect to x
When to use it
To differentiate any power of x, term by term: polynomials, roots (√x = x^½) and reciprocals (1/x² = x⁻²). Rewrite roots and fractions as powers first.
Worked example
Differentiate \(y=6\sqrt{x}-\dfrac{4}{x^2}\).
- Write as powers: \(y=6x^{1/2}-4x^{-2}\)
- \(\dfrac{dy}{dx}=3x^{-1/2}+8x^{-3}\)
Answer: \(\dfrac{dy}{dx}=\dfrac{3}{\sqrt{x}}+\dfrac{8}{x^3}\)
Common mistake
Differentiating 1/x² as 1/(2x). Write it as x⁻² first: the derivative is −2x⁻³. And the derivative of a constant is 0.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Differentiation
- Revise IAL P1 differentiation
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the power rule for differentiation formula?
The power rule for differentiation is d/dx (xⁿ) = nxⁿ⁻¹, and d/dx (axⁿ) = anxⁿ⁻¹. n: the power: any number, including fractions and negatives; a: a constant multiple; d/dx: differentiate with respect to x.
Is the power rule (differentiation) given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
How do I differentiate √x or 1/x?
Write them as powers: √x = x^½ gives ½x^(−½) = 1/(2√x), and 1/x = x⁻¹ gives −x⁻² = −1/x².
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.