# What is the derivative of e^(x/2)? **URL:** \ **Category:** Factual Questions\ **Created:** [February 19, 2012, 8:53pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351 "2012-02-19T20:53:46Z")\ **Posts on this page:** 13\ **Page:** 1
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [February 19, 2012, 8:53pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/1 "2012-02-19T20:53:46Z")
Not homework; just trying to remember how. ∫e[sup]x/2[/sup]\*dx IIRC, you take the derivative of the ‘inner’ function (x/2) and multiply it by the derivative of the ‘outer’ function (e[sup]x/2[/sup]). So the derivative of the ‘inner’ function is (1/2) \* x[sup]0[/sup] = (1/2) \* 1 = 1/2 and the derivative of the ‘outer’ function is e[sup]x/2[/sup], and the derivative of e[sup]x/2[/sup] is (1/2)\*e[sup]x/2[/sup], right? ---
**Author:** ![Simplicio](https://avatars.discourse-cdn.com/v4/letter/s/c37758/32.png) [@Simplicio](https://boards.straightdope.com/u/Simplicio)\ **Post date:** [February 19, 2012, 9:01pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/2 "2012-02-19T21:01:25Z")
FYI: Your words say derivitive, but I get an integral symbol on my computer. But otherwise your right. d/dx e^(x/2) = .5\*e^(x/2) ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [February 19, 2012, 9:08pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/3 "2012-02-19T21:08:58Z")
> [@Simplicio](#): > > FYI: Your words say derivitive, but I get an integral symbol on my computer. > > But otherwise your right. d/dx e^(x/2) = .5\*e^(x/2) Sorry, I meant integral. Thanks. ---
**Author:** ![cerberus](https://avatars.discourse-cdn.com/v4/letter/c/71e660/32.png) [@cerberus](https://boards.straightdope.com/u/cerberus)\ **Post date:** [February 19, 2012, 9:09pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/4 "2012-02-19T21:09:37Z")
d exp(f(x))/ dx = (d(f(x))/dx) exp(f(x)) ---
**Author:** ![Simplicio](https://avatars.discourse-cdn.com/v4/letter/s/c37758/32.png) [@Simplicio](https://boards.straightdope.com/u/Simplicio)\ **Post date:** [February 19, 2012, 9:18pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/5 "2012-02-19T21:18:40Z")
> [@Johnny\_L.A](#): > > Sorry, I meant integral Then your answer is wrong. The derivative is half the original function, the integral is twice the original function. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [February 19, 2012, 9:19pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/6 "2012-02-19T21:19:18Z")
> [@Simplicio](#): > > FYI: Your words say derivitive, but I get an integral symbol on my computer. > > But otherwise your right. d/dx e^(x/2) = .5\*e^(x/2) > [@Johnny\_L.A](#): > > Sorry, I meant integral. > > Thanks. You’re right that the **derivative** of e[sup]x/2[/sup] is (1/2)e[sup]x/2[/sup]. But that means that the indefinite **integral** , or antiderivative, of (1/2)e[sup]x/2[/sup] is e[sup]x/2[/sup] + C, and then the integral of e[sup]x/2[/sup] is 1/(1/2)e[sup]x/2[/sup] + C, or 2e[sup]x/2[/sup] + C. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [February 19, 2012, 9:22pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/7 "2012-02-19T21:22:31Z")
[Let me WolframAlpha that for you.](http://www.wolframalpha.com/input/?i=integrate+e%5E%28x%2F2%29) ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [February 19, 2012, 10:06pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/8 "2012-02-19T22:06:07Z")
> [@Thudlow\_Boink](#): > > then the integral of e[sup]x/2[/sup] is 1/(1/2)e[sup]x/2[/sup] + C, or 2e[sup]x/2[/sup] + C. OK, how do you get from ∫e[sup]x/2[/sup] to 2e[sup]x/2[/sup] + C? (I know where the C comes from, since I didn’t include limits.) Wait, let me think… If the coefficient of e is one, then to get the derivative 1/2 the coefficient of e needs to be 2. But how do I get there logically? Also: Why do I get these things in my head and spend time on them? ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [February 19, 2012, 10:16pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/9 "2012-02-19T22:16:26Z")
> [@Johnny\_L.A](#): > > OK, how do you get from ∫e[sup]x/2[/sup] to 2e[sup]x/2[/sup] + C? (I know where the C comes from, since I didn’t include limits.) > > Wait, let me think… If the coefficient of e is one, then to get the derivative 1/2 the coefficient of e needs to be 2. But how do I get there logically? > > Also: Why do I get these things in my head and spend time on them? u-substitution works: ∫e[sup]x/2[/sup] dx = ∫e[sup]x/2[/sup] 2 d(x/2) = 2 ∫e[sup]u[/sup] du = 2 e[sup]u[/sup] + c = 2e[sup]x/2[/sup] + c ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [February 19, 2012, 10:26pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/10 "2012-02-19T22:26:11Z")
U-substitution! :smack: I remember that! Thanks. ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [February 19, 2012, 10:49pm UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/11 "2012-02-19T22:49:23Z")
Did I mention I suck at math? Oh, yes. In several threads. 😃 And yet… I can’t help needlessly torturing myself. :rolleyes: ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [February 20, 2012, 3:05am UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/12 "2012-02-20T03:05:01Z")
> [@Johnny\_L.A](#): > > Did I mention I suck at math? Oh, yes. In several threads. 😃 > > And yet… I can’t help needlessly torturing myself. :rolleyes: Only on this board would someone working calculus problems think that they suck at math. 🙂 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [February 20, 2012, 5:15am UTC](https://boards.straightdope.com/t/what-is-the-derivative-of-e-x-2/613351/13 "2012-02-20T05:15:03Z")
The rules for derivatives are all well-established and algorithmic, such that any standard function or combination of standard functions has a derivative that is a combination of standard functions, and that combination can be found straightforwardly. The same is not, however, true of integrals. So when one is presented with a pair of functions and the claim that the one is the integral of the other, the simple way to check this claim is to perform the derivative, not to perform the integral.