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Topic: Proof of derivative of ln x or e^x ?
Replies: 63   Last Post: Dec 20, 2001 12:23 PM

 Messages: [ Previous | Next ]
 Herman Rubin Posts: 6,721 Registered: 12/4/04
Re: Proof of derivative of ln x or e^x ?
Posted: Dec 18, 2001 8:07 AM

In article <9vjhr9\$dgh\$1@pc18.math.umbc.edu>,
Rouben Rostamian <rouben@math.umbc.edu> wrote:
>In article <wrameyxiii-01573F.16031316122001@netnews.attbi.com>,

>>In article <9vhg41\$a0r\$1@pc18.math.umbc.edu>,
>> rouben@math.umbc.edu (Rouben Rostamian) wrote:

>>> But that's beside the point. By defining log x as the inverse
>>> function of 10^x or e^x you are just postponing the problem. How
>>> do you define 10^x? Don't tell me that's defined in terms of
>>> logarithms :-)

I doubt that when Napier invented logarithms he did not
know the meaning of a^x for a positive and x real. He
might not have used the epsilon-delta notation, but he
certainly had the idea. Tabulation and interpolation were
the methods used BC (before calculus).

I recall having read that logarithms to base 10 were first
calculated by computing 10^x for x of the form m/2^n and
inverting. Also, Napier first computed logarithms to the
base .9999999 by multiplication.

>>> After you clean up your definition of 10^x, stand back and
>>> see if that is any more appealing than the definition of ln x
>>> as the antiderivative of 1/x.

>>I've always found it humorous that calculus texts start getting sweaty with
>>rigor on the derivatives of the exponential functions b^x (b > 0) and their
>>companions, the logarithms. Do these students even know, with rigor, what
>>sqrt(x) is? Usually not, but they've been merrily differentiating it for
>>weeks.

That students do not know the meaning of powers and roots
is abysmal. But with the current emphasis on calculation,
not on understanding, this is not that surprising.
Mathematical concepts need to be taught early, NOT after
computation. The students who learn how to carry out
calculations of derivatives and antiderivatives are not in
any better position to understand what they mean than they
were initially, and I believe they are in a worse shape.

>I agree with your assessment of the situation. A good part of the
>fine points of calculus is lost on the majority of students.

Of course they are lost. If the concepts of limit and
derivative are "done" in one lecture, with one homework
assignment, and never appear on final examinations, what do
you expect? The impression is given that it is not of much
importance. Rather, it is the drill in calculation which
should be considered of little importance, from elementary
school on. It is useless to know how to add if one does
not know what addition means. They do not know even what
the properties of the integers are when they take calculus.

...............

>Back to the original point of this thread: I don't want to give the
>incorrect impression that I am rigidly set against defining ln x as
>the inverse function of e^x.

This is how it was done from around 1600 until almost the present.

It wouldn't hold it against a calculus
>book that does. I will be even happier if both approaches -- i.e.
>defining e^x in terms of ln x and visa versa -- were presented, possibly
>one version in the main text and the other version as an exercise.

One problem with starting with the logarithm as an integral
is the problem of handling numbers less than 1. On the
other hand, showing that (1+x/n)^n is increasing in n for x
> 0 by using the binomial theorem and termwise comparison,
and proceeding from there by using properties of limits and
infinite series, is, I believe, quite within the range of
anyone who has had a modicum of high school algebra and can
think. There is no point of delaying it until later.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

Date Subject Author
12/14/01 Sam Lachterman
12/14/01 David C. Ullrich
12/14/01 Ronald Bruck
12/14/01 Dann Corbit
12/15/01 markov@moreland.com
12/14/01 Dave L. Renfro
12/14/01 Tomasu
12/14/01 Simon S. Goldenberg
12/14/01 Sam Lachterman
12/15/01 Daniel R. Grayson
12/17/01 David C. Ullrich
12/17/01 Denis Feldmann
12/17/01 Daniel R. Grayson
12/18/01 David C. Ullrich
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12/17/01 The Scarlet Manuka
12/14/01 Martin Green
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12/15/01 Martin Green
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12/15/01 Martin Green
12/16/01 marsh@agora.rdrop.com
12/17/01 David C. Ullrich
12/20/01 Denis Feldmann
12/20/01 Mark Thakkar
12/20/01 Denis Feldmann
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12/15/01 Dave Rusin
12/15/01 Simon S. Goldenberg
12/15/01 Jon and Mary Frances Miller
12/15/01 Severian
12/17/01 Randy Poe
12/17/01 The Scarlet Manuka
12/15/01 Tom Ace
12/15/01 Darrell
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12/15/01 Severian
12/15/01 N.Bubis
12/15/01 Severian
12/15/01 Ronald Bruck
12/16/01 marsh@agora.rdrop.com
12/19/01 Ronald Bruck
12/19/01 Herman Rubin
12/16/01 Richard Carr
12/19/01 Ronald Bruck
12/15/01 onand@usa.net
12/15/01 marsh@agora.rdrop.com
12/15/01 G.E. Ivey
12/15/01 Rouben Rostamian
12/16/01 marsh@agora.rdrop.com
12/16/01 Rouben Rostamian
12/16/01 marsh@agora.rdrop.com