Here’s another way of doing it:
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.
- A Sparse Online Update for a Hierarchical Model (machinedlearnings.com)
2 responses to “An alternative proof for the derivative of ln(x)”
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.
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
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.