# Yet Another Calculus Question **URL:** \ **Category:** Factual Questions\ **Created:** [January 30, 2003, 3:11am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514 "2003-01-30T03:11:32Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [January 30, 2003, 3:11am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/1 "2003-01-30T03:11:32Z")
Last one, I promise. I can’t find any resource that gives me the identity for the derivative of e raised to the power of X if X itself is subjected to an exponential operation. Now, we know that where Y=e^X the derivative is Y’=e^X - weird, but true. And we know that where Y=e^KX, then Y’=Ke^KX. But what if Y=e^sin(X) or Y=e^(X^2) ? ---
**Author:** ![Super\_Gnat](https://avatars.discourse-cdn.com/v4/letter/s/e274bd/32.png) [@Super\_Gnat](https://boards.straightdope.com/u/Super_Gnat)\ **Post date:** [January 30, 2003, 3:16am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/2 "2003-01-30T03:16:56Z")
It’s chain rule, right? f’r’instance y=e^x[sup]2[/sup] then y’=2xe^x[sup]2[/sup] and y"=2x(2xe^x[sup]2[/sup])+2(e^x[sup]2[/sup]) ---
**Author:** ![js\_africanus](https://avatars.discourse-cdn.com/v4/letter/j/87869e/32.png) [@js\_africanus](https://boards.straightdope.com/u/js_africanus)\ **Post date:** [January 30, 2003, 3:20am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/3 "2003-01-30T03:20:11Z")
> [@](#): > > But what if Y=e^sin(X) or Y=e^(X^2) ? I think you just follow the chain rule. Take the deriviative of the thing in the parentheses, then the next thing and so-on. Then multiply them all together. Here’s a place to start. ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [January 30, 2003, 3:21am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/4 "2003-01-30T03:21:10Z")
Yes, basically: y = e^f(x) dy = e^f(x) d(f(x)) which you see as y = e^x dy = e^x dx when f(x) is x. Therefore, y=e^sin(x) dy = cos(x) e^sin(x) dx to use the example Super Gnat didn’t. Ordinary chain rule. 🙂 ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [January 30, 2003, 3:21am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/5 "2003-01-30T03:21:29Z")
And for y = e[sup]sin(x)[/sup] y’=cos(x) e[sup]sin(x)[/sup] ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [January 30, 2003, 4:39am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/6 "2003-01-30T04:39:26Z")
Like everyone else said, it’s the chain rule. ---
**Author:** ![RickJay](https://avatars.discourse-cdn.com/v4/letter/r/bb73d2/32.png) [@RickJay](https://boards.straightdope.com/u/RickJay)\ **Post date:** [January 30, 2003, 5:17am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/7 "2003-01-30T05:17:31Z")
Beee-zarre. ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [January 30, 2003, 6:58am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/8 "2003-01-30T06:58:23Z")
Rickj: y=e^x, y’=e^x is not weird. It’s not like it’s luck or something. It is defined that way. The natural log is defined as the solution to the the equation: F(x) = F’(x). ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 30, 2003, 11:02am UTC](https://boards.straightdope.com/t/yet-another-calculus-question/151514/9 "2003-01-30T11:02:57Z")
> [@](#): > > \*Originally posted by John Mace \* > \*\*Rickj: > > y=e^x, y’=e^x is not weird. It’s not like it’s luck or something. It is defined that way. The natural log is defined as the solution to the the equation: F(x) = F’(x). \*\* I think he meant to day that the exponential is defined by F’ = F. The natural log is defined as the solution to f’(x) = 1/x. More to the point, one can prove that if f(x) = a^x, then f’(x) is proportional to f(x) and e is defined as the unique number for which the proportionality constant is 1.