## An alternative proof for the derivative of ln(x)

Here’s another way of doing it:

$\frac{d}{dy}log_e(x) = \lim_{\delta \to 0} \frac{log_e(x+\delta) - log_e(x)}{\delta}$

As, $log_e(x+\delta) - log_e(x) = log_e(1 +\frac{\delta}{x})$

So, $\frac{d}{dy}log_e(x) = \lim_{\delta \to 0} log_e((1 + \frac{\delta}{x})^{\frac{1}{\delta}})$

Let $\gamma = \frac{\delta}{x}$, then

$\frac{d}{dy}log_e(x) = \lim_{\gamma \to 0} log_e((1 + \gamma)^{\frac{1}{x\gamma}}) = \lim_{\gamma \to 0}\frac{1}{x}log_e((1 + \gamma)^{\frac{1}{\gamma}})$

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

$e = \lim_{n \to 0}(1 + n)^{\frac{1}{n}}$

Hence, $\lim_{\gamma \to 0}\frac{1}{x}log_e((1 + \gamma)^{\frac{1}{\gamma}}) = \frac{1}{x}log_e(e) = \frac{1}{x}$ and so $\frac{d}{dx}log_e(x) = \frac{1}{x}$.

More or less cribbed from here.

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### 2 responses to “An alternative proof for the derivative of ln(x)”

1. @Our_Frank

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!

2. 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.