1. ## Problem with integration

$f(x)= \left( 1-\frac{x}{2} \right)^{\frac{a}{x}}$

How can i find f ' (0) ?

2. Originally Posted by atlantis84
$f(x)= \left( 1-\frac{x}{2} \right)^{\frac{a}{x}}$

How can i find f ' (0) ?
First find f(0). Take ln's so

$\ln f = \frac{a \ln \left(1 - \frac{x}{2}\right)}{x}$ (1)

use L'Hopital's rule. This gives $f(0) = e^{-a/2}.$ Then differentiate (1) and use L'Hoptials rule again on this. This gives

$
\frac{f'(0)}{f(0)} = - \frac{a}{8}
$

so $f'(0) = - \frac{a}{8} e^{-a/2}.$

3. Thnxxx for helping, i got it till $f(0) = e^{-a/2}$

but i have a really hard time calculating

$
\frac{f'(0)}{f(0)} = - \frac{a}{8}
$

Could you give me the exact calculations? thnx anyway