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 \(y=2x^5-3x^{-2}+4\) with respect to \(x\).
- \(\dfrac{d}{dx}(2x^5)=10x^4\), \(\dfrac{d}{dx}(-3x^{-2})=6x^{-3}\), \(\dfrac{d}{dx}(4)=0\)
Answer: \(\dfrac{dy}{dx}=10x^4+\dfrac{6}{x^3}\)
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 |
|---|---|
| Class 11 | Learn it: CBSE gives no formula booklet |
| Class 12 | Learn it: CBSE gives no formula booklet |
From our own CBSE Maths formula sheets, in our words.
Practise and revise
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?
Class 11: learn it: CBSE gives no formula booklet. Class 12: learn it: CBSE gives no formula booklet. This comes from our own CBSE 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 CBSE Math Revision. Not produced or endorsed by CBSE or NCERT.