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