Derivative of e^x

The number $e$ is chosen so that the function is its own derivative. That is why the derivative of $e^x$ is $e^x$.

For a base $a$ other than $e$, $a^x$ picks up a factor of $\ln a$. Because exponentials keep their shape under differentiation, they are a common training ground for the chain rule.

Examples

1. 01Differentiate$\displaystyle \frac{d}{dx} e^x$AnswerHide

   Answer$\displaystyle e^x$
2. 02Differentiate$\displaystyle \frac{d}{dx} e^{2x}$AnswerHide

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

   Answer$\displaystyle -e^{-x}$
4. 04Differentiate$\displaystyle \frac{d}{dx} 3e^{x}$AnswerHide

   Answer$\displaystyle 3e^{x}$
5. 05Differentiate$\displaystyle \frac{d}{dx} e^{x^{2}}$AnswerHide

   Answer$\displaystyle 2xe^{x^{2}}$
6. 06Differentiate$\displaystyle \frac{d}{dx} 2^{x}$AnswerHide

   Answer$\displaystyle 2^{x}\log 2$

FAQ

Q1Does $e^x$ stay the same no matter how many times you differentiate?Show answerHide

A

Yes. Every derivative is $e^x$. If there is something inside, like $e^{2x}$, each derivative multiplies by the derivative of the inside, 2.

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

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