is it possible to demonstrate the derivative of log (log x) using only limits ? i.e., can it be shown what the derivative of log (log x) is by setting up the limit:

limit as h tends to 0 of [ log (log (x + h)) - log (log (x))] / h

Question

is it possible to demonstrate the derivative of log (log x) using only limits ?  i.e., can it be shown what the derivative of log (log x) is by setting up the limit:

limit as h tends to 0 of [ log (log (x + h)) - log (log (x))] / h

anonymous · Answer

$$\lim_{h \rightarrow 0} \frac{ \log (\log (x + h)) - \log (\log x) }{ h }$$

anonymous · Answer

i tried this on paper several times with no luck.  i was wondering if anyone else could figure it out

parthkohli · Answer

$$\log(\log(x +h)) - \log(\log(x)) = \log\left(\dfrac{\log{(x +h)}}{\log(x)}\right)$$assume $k = x + h$ so $k - x = h$.$$\lim_{k\to x} \log\left(\dfrac{\log(k)}{\log(x)}\right) = \log\left(1\right) = 0$$

parthkohli · Answer

Is that the right answer?

parthkohli · Answer

No, I guess not. Sorry

parthkohli · Answer

Oh wait, that is incomplete.

parthkohli · Answer

$$\lim _{k \to x} \dfrac{\left(\log\left(\log(k) \over \log(x)\right)\right)}{k-x}$$