# Calculating d/dx e^x from first prinicples
**URL:** https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874
**Category:** Factual Questions
**Created:** [December 16, 2016, 5:57pm UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874 "2016-12-16T17:57:34Z")
**Posts on this page:** 7
**Page:** 2
### Author:  [@glowacks](https://boards.straightdope.com/u/glowacks)
#### Post date: [December 24, 2016, 4:43am UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/21 "2016-12-24T04:43:47Z")
> [@Dr.Strangelove](#):
>
> It’s just the first step in their proof: e = lim[n-\>inf] (1 + 1/n)^n
No, I don’t mean what e itself is. What e^x means, since exponentiation is only naively defined for rational x. The question is “what does it mean when you use a random number that may be irrational as an exponent?” You need to answer this question if you want to talk about the derivative, since those only work on continuous functions. Without knowing how to obtain what e^x is for irrational x, it’s pointless to ask what its derivative is.
> [@septimus](#):
>
> h is the formal argument of lim (what is this type of variable called?) but is also being used outside the scope of the lim. I see that you get the right answer, but is there some comment needed to justify this step?
I had the same sort of objection (with less specification exactly where). Whoever did that proof clearly never did any actual upper-level analysis. It works out because everything is well behaved, but of course, you don’t technically know that when you’re working from first principles.
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### Author:  [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)
#### Post date: [December 24, 2016, 7:17pm UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/22 "2016-12-24T19:17:56Z")
> [@septimus](#):
>
> e[sup]x[/sup] and ln x are the inverses of each other, so whichever derivative you prove first, you get the other one almost for free, via a form of the chain rule:
>
> f\*\*‘**(y) = 1 / (f[sup]-1[/sup])**’\*\*(f(y))
>
> h is the formal argument of _lim_ (what is this type of variable called?) but is also being used _outside the scope_ of the _lim_. I see that you get the right answer, but is there some comment needed to justify this step?
There sure is, see my previous post.
Let me clarify my point. I don’t mean that you have to go through all the steps I outlined in elementary calculus. You could not get through an elementary calculus course that way. What I mean is that there are certain things you avoid in elementary calculus. For example, what is a real number? You might say it is an infinite decimal\*, but do you know how to add, worse to multiply, infinite decimals. In ordinary arithmetic you start from the right hand end and add with carry. But infinite decimals have no right hand end. You might try dealing with the carries involved with adding .999999… to itself to get a picture of the problem involved. What you have to do is add all the finite approximations and prove they converge. Yes, it can be done, but not in elementary calc course. So you take the real numbers as a given and assume that arithmetic is what it seems. But in that case, you might as well take the exponential as given by the power series in which case it is obviously its own derivative.
\*There are other possible definitions, whose arithmetic is more tractable, but none is elementary. Probably Cauchy sequences is the simplest, but there you into convergence questions.
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### Author:  [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)
#### Post date: [December 24, 2016, 9:33pm UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/23 "2016-12-24T21:33:15Z")
> [@glowacks](#):
>
> No, I don’t mean what e itself is. What e^x means, since exponentiation is only naively defined for rational x. The question is “what does it mean when you use a random number that may be irrational as an exponent?” You need to answer this question if you want to talk about the derivative, since those only work on continuous functions.
Well, ok. But how far back do you want to go? I don’t think we want to build up all of math from scrath here.
In any case, it seems straightforward, and we proceed similarly to the construction of the reals. To construct an irrational real, we can identify it as the “cut” between two sets: one set consisting of all the rationals below our target number, and the other set containing all rationals above the number. There is a unique number that gets “squeezed out” between the two sets and that is the new irrational.
Likewise, we can say that n^x for irrational x is the number that gets squeezed out between two sets: one of which is all n^a where a is a rational \x. Because exponentiation is monotonic, the new number satisfies n^a\
### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [December 25, 2016, 4:51am UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/24 "2016-12-25T04:51:46Z")
Yeah, that method works, too. But of course, one of the more straightforward ways to prove that exponentiation thus defined works is to produce a sample function and then prove that your sample function meets both the definition and the desired properties.
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### Author:  [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)
#### Post date: [December 26, 2016, 9:44am UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/25 "2016-12-26T09:44:22Z")
No doubt about that. All else being equal, I do favor the calculus-centered approach of defining e[sup]x[/sup] as the function which is its own derivative, and then proving that the function (and generalization of the exponential function) has all the properties we like.
But as usual in math, there’s more than one way to do it, and there’s enlightenment to be had in examining the different approaches. Not to mention a certain perverse enjoyment in doing things the hard way. Math is, above all, a _game_, and games should be fun.
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### Author:  [@Manlob](https://boards.straightdope.com/u/Manlob)
#### Post date: [December 26, 2016, 3:44pm UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/26 "2016-12-26T15:44:11Z")
The function that equals its own derivative is not unique.
e^x
7_e^x
3_e^(x-4.3)
…
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### Author:  [@OldGuy](https://boards.straightdope.com/u/OldGuy)
#### Post date: [December 26, 2016, 7:49pm UTC](https://boards.straightdope.com/t/calculating-d-dx-e-x-from-first-prinicples/774874/27 "2016-12-26T19:49:16Z")
> [@Manlob](#):
>
> The function that equals its own derivative is not unique. …
That’s because as with any differential equation you need a boundary condition to determine it completely. So f(x) = e^x is the function whose derivative is equal to itself and satisfies f(0) = 1.
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