Product rule

The product rule is: differentiate one factor at a time and add. Differentiate the first and keep the second, plus keep the first and differentiate the second.

If you can expand first, expanding is sometimes faster.

Examples

1. 01Differentiate$\displaystyle \frac{d}{dx}\,((x+1)(x^{2}+2))$AnswerHide

   Answer$\displaystyle 3x^{2}+2x+2$
2. 02Differentiate$\displaystyle \frac{d}{dx}\,(x^{2}(x^{3}-1))$AnswerHide

   Answer$\displaystyle 5x^{4}-2x$
3. 03Differentiate$\displaystyle \frac{d}{dx}\,((2x-1)(x^{2}+x))$AnswerHide

   Answer$\displaystyle 6x^{2}+2x-1$
4. 04Differentiate$\displaystyle \frac{d}{dx}\, x^{2}\sin x$AnswerHide

   Answer$\displaystyle 2x\sin x + x^{2}\cos x$
5. 05Differentiate$\displaystyle \frac{d}{dx}\, xe^{x}$AnswerHide

   Answer$\displaystyle (x+1)e^{x}$
6. 06Differentiate$\displaystyle \frac{d}{dx}\, x^{2}\log x$AnswerHide

   Answer$\displaystyle 2x\log x + x$

FAQ

Q1Do the product rule and expanding first give the same result?Show answerHide

A

Yes. For two polynomials, expanding is often faster. When $e^x$ or $\sin x$ is involved, you need the product rule.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05