Prove that d/dx lnx = 1/x WITHOUT assuming it or using the fact d/dx e^x = e^x @amistre64 (I used to know the good proof but I forgot)

Question

amistre64 · Answer

ln(x+h) - ln(x)
-------------
         h

anonymous · Answer

-.-

amistre64 · Answer

or is the other fashion better for this:

lnb - lna
-------
  b-a

zarkon · Answer

what is your definition of $\ln(x)$

anonymous · Answer

Use the proof that uses the limit definition of E.

amistre64 · Answer

limit defintion of epsilon deltas?

anonymous · Answer

$$e = \lim_{n \rightarrow \infty}(1+\frac{ 1 }{ n })^{n}$$

amistre64 · Answer

i dont think i know of a proof that would use that off hand.

anonymous · Answer

Oh another question, how come e is 2.718 when the limit definition just brings a 1 as the answer?

amistre64 · Answer

squeeze thrm plays a part i believe

anonymous · Answer

Because of n is infinity then 1/n is 0 and 1^n is 1for all n

zarkon · Answer

you don't get to dictate how fast each individual n goes to infinity....they all go to infinity at the same rate

amistre64 · Answer

yeah ... that logic :)

((n+1)/n)^n

(2/1)^1 = 2
(3/2)^2 = 2.25
(4/3)^3 = 2.370370...
(5/4)^4 = 2.441406...
etc
(101/100)^100 = 2.7048...

the limit of the sequence evens out to e