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could someone explain to me just how the integral of x^-1 is ln(x) ? cheers Saucers ps, im not looking for something like "cos ln is the derivitive of x^-1"

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Well the reason you don''t want us to give is the fundamental theorem of calculus which is pretty stupid to ignore =p

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http://www.mathstat.uottawa.ca/Profs/Rossmann/MAT1320A_files/3-7_Log.pdf

[edited by - cozman on February 20, 2004 10:04:12 PM]

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just like the op inferred, you''ll need to understand the fundamental theorem of calculus, and then you''ll realize the inverse relationship that exists between derivatives and integrals. So then if you understood that the derivative of ln(x) is 1/x, then going backwards from it will make sense.

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If I remember right (it''s been a while), ln(n) is defined as being the integral of 1/x. Or more specifically, the integral of 1/x from 1 to n, I believe. So there''s not really as much mathematical derivation involved. It''s just defined that way. Whenever you need to calculate the natural log of a number, and you don''t have a premade chart made, then you have to approximate the integral, since that is precisely how it is defined.

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Uhhh. Ok. Right. Lemme dig out the calculus book. I figured someone would have posted this by now, but since nobody is... here it goes...

(slthough cozman posted something that is good. you should look at it.)

d/dx(loga x) = 1/(xlna)

Proof:
ay=x
Differentiate implicitly

ay(ln a)(dy/dx)=1

dy/dx=1/ayln a = 1/(x*ln a)

If a=e,
d/dx(ln x)=1/x

That''s the way from function to derivative.

Now I''ll use my pat answer and say ... "the derivative of 1/x is ln x" ... and hope you understand. ;-) I suppose if I get bored tonight i''ll do a derivation with reimann integrals, but i think that it''d be really kinda ugly.

Scout

All polynomials are funny - some to a higher degree.
Furthermore, polynomials of degree zero are constantly funny.

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Another nice way to show it:

if g is the inverse of f, and x is a real number in the domain of G then: Dg(x) = 1/Df(g(x))
and f(g(x)) = x

Not hard to prove, but using this:

g(x) = ln(x)
f(x) = exp(x)

ln(x) = 1/exp(ln(x)) = 1/x

Use the fundamental theroem of calculus to go backwards.

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hey, thanks everyone, that really helped :D

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When you have a derivative over the original function (1 is the derivative of x), the anti-derivative of that function is always ln(denominator) ex: dx/dy = 2x/(x^2) y = ln(x^2). Because to take the derivative of an ln() function, take the derivative of what''s in the parentheses and put it ''on top'' of what''s in the parentheses.

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