Re: Derivative of exponential function

From: Ian Taylor (iantaylor2uk_at_yahoo.co.uk)
Date: 10/08/04 

Date: 8 Oct 2004 11:06:27 -0700

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. 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

Relevant Pages

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