Re: Derivative of exponential function

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 10/09/04 

Date: Sat, 09 Oct 2004 06:36:36 -0500

On 8 Oct 2004 11:06:27 -0700, iantaylor2uk@yahoo.co.uk (Ian Taylor)
wrote:

>mstemper@siemens-emis.com (Michael Stemper) wrote in message news:<200410071712.i97HCAL62738@mickey.empros.com>...
>> I'm in the process of home-schooling my son in calculus. We're about to
>> finish up on limits, so that we can start differentiation. I've relearned
>> enough that I've been able to derive, from the definition of derivative,
>> the derivatives of c, x, cx, f(x)+g(x), f(x)*g(x), f(g(x)), sin(x),
>> cos(x). From those, I can get the derivatives of f(x)-g(x), f(x)/g(x),
>> (f(x))^n, (f(x))^(-n), tan(x). A pretty solid beginning. But there's
>> one big hole: I can't prove that the derivative of e^x is e^x.
>>
>> I can see that if Lim(h->0) ((e^h-1)/h) == 1, then I'm set. Numerically,
>> I can crank out values for ((e^h-1)/h) with h getting very small, and
>> see that they get really close to 1. But, that's not proof.
>>
>> I was going to base a proof on the Taylor series for e^x, but that
>> depends upon already knowing the nth derivatives of e^x, so that was
>> out. Then I was going to try using my knowledge of the behavior of
>> e^x, but upon examination, that was all based on knowing things about
>> its derivative.
>>
>> Is there some simple (or subtle) trick that I'm overlooking? Is the
>> proof of this limit actually incredibly hard?
>>
>> I can't look in the book, because it's with him (he lives with his
>> mother). I've even tried typing "((e^h-1)/h)" into Google, but that
>> just turned up a bunch of PDF files. Any help, or am I going to need
>> to do some serious hand-waving?
>
>How about this - define a function f(x) to be the same as it's
>derivative.

With the condition f(0) = 1 that is indeed a way to define
the exponential, and it's fairly elegant. But to make it an
actual valid definition you first need to know something
about existence and uniqueness for differential equations.

>Assume you can do a series expansion of f(x). Explicitly
>do the series expansion for f(x) and d/dx (f(x)) and set them equal to
>each other (for all x). This should define the coefficients in the
>series expansion. Now call this function f(x) the exponential
>function, exp(x).
>
>I have seen this done before and thought it was quite neat. I don't
>know if it can mathematically justified, as I am only a physicist, but
>it looks quite neat to me, and you can avoid all the handwaving.
>
>Ian Taylor

************************

David C. Ullrich

Relevant Pages

  • Re: Derivative of exponential function
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  • Re: Derivative of exponential function
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