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
\(f(x)=4x^3-5x^2+7x-2\). Find \(f'(x)\) and the gradient of the curve at \(x=2\).
- \(f'(x)=12x^2-10x+7\)
- \(f'(2)=12(4)-10(2)+7=48-20+7\)
Answer: \(f'(x)=12x^2-10x+7\); gradient at \(x=2\) is \(35\)
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 |
|---|---|
| AA SL | In the IB formula booklet |
| AI SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA SL rules of differentiation questions
- Practise AI SL differentiation rules questions
- Revise Differentiation (AA SL)
- Revise Differentiation (AI SL)
- Print AA SL 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?
AA SL: in the IB formula booklet. AI SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.