Chain rule

Differentiate the outer function while leaving the inside as it is, then multiply by the derivative of the inside.

Forgetting that inner factor is the classic mistake.

Examples

1. 01Differentiate$\displaystyle \frac{d}{dx}\,\cos(5 x^{5})$AnswerHide

   Answer$\displaystyle -25x^{4} \sin(5 x^{5})$
2. 02Differentiate$\displaystyle \frac{d}{dx}\,\log_e(x^{2})$AnswerHide

   Answer$\displaystyle \frac{2}{x}$
3. 03Differentiate$\displaystyle \frac{d}{dx}\,e^{2 x^{2}}$AnswerHide

   Answer$\displaystyle 4x e^{2 x^{2}}$
4. 04Differentiate$\displaystyle \frac{d}{dx}\,e^{x^{4}}$AnswerHide

   Answer$\displaystyle 4x^{3} e^{x^{4}}$
5. 05Differentiate$\displaystyle \frac{d}{dx}\,(3 \cos x)^{3}$AnswerHide

   Answer$\displaystyle -9 (3 \cos x)^{2} \sin x$
6. 06Differentiate$\displaystyle \frac{d}{dx}\,(\log_e x)^{5}$AnswerHide

   Answer$\displaystyle \frac{5 (\log_e x)^{4}}{x}$

FAQ

Q1What is easy to forget with the chain rule?Show answerHide

A

Multiplying by the derivative of the inside. For $(2x+1)^5$, take $5(2x+1)^4$ and multiply by 2 to get $10(2x+1)^4$.

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

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