cartesian product … stuff about computing mostly

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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Adrian McMenamin

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

  1. @Our_Frank Avatar
    @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. Adrian McMenamin Avatar
    Adrian McMenamin

    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.

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