Here’s another way of doing it:

As,

So,

Let , then

Now, one of the properties of , Euler’s number, is that it is the limit of two mathematical expressions, one of which is:

Hence, and so .

More or less cribbed from here.

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Very interesting stuff.

You got me trawling through some long unvisited corridors of memory anyway – it’s been

a while since I’ve thought about this stuff in any detail.

I think possibly the reason that you found few proofs of the derivative of the

natural logarithm on the web is that (by my understanding anyway) the derivative

of ln(x) is 1/x by definition – or at least, the function ln(x) is defined as the

integral of 1/x (between 1 and x). Therefore, the derivative of

ln(x) is 1/x, by the original integral definition of the function.

By investigating the properties of this integral you can demonstrate that it is a

logarithmic function and that its base is Euler’s number, e.

A wonderful result! – given that the integral of 1/x is undefined by the standard

definition of polynomial integrals i.e. integral of x^n = (x^(n+1))/(n+1)

I had fun – trawling those corridors of memory. I’ve posted a “proof” from the

perspective of the integral definition of ln(x) here if you’re interested.

http://ourfrank.blogspot.com/2011/12/musings-on-eulers-number-and-natural.html

Interesting because I find it’s often tricky to be quite sure with “proofs”

what’s a priori knowledge.

I suspect (thought I’m not entirely sure) that both of the proofs posted here are

implicitly tautological – i.e. require the derivative of ln(x) to be defined as 1/x

already.

In the first, for example, I suspect that the derivative of ln(x) needs to

have been defined to be 1/x to be able show that the derivative of exp(x) is exp(x).

And in the second, it looks like the derivative of ln(x) needs to

have been defined to be 1/x in order to show

that “e equals the limit of the expression ‘1 plus 1 divided by x'”.

See, for example, http://users.rcn.com/mwhitney.massed/defn_of_e/defn_of_e.html

Anyway – hope that’s helpful/interesting/useful in some way.

Best regards – and Happy Christmas!

Thanks – as I said in an earlier post I too am remembering this from days gone by. Started because I wanted to understand queuing simulations better – as still thinking about memory management issues.