Question #2207

If f(x)=(1/x)[sup]x[/sup] , find f ``(1)

Expert's answer

<img src="/cgi-bin/mimetex.cgi?f%28x%29%20=%20%5Cleft%20%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%20%29%5Ex%20=e%5E%7B-x%20%5Cln%7Bx%7D%7D%20%5C%5C%20f%27%28x%29%20=%20%28-%5Cln%7Bx%7D-1%29e%5E%7B-x%20%5Cln%7Bx%7D%7D%20%5C%5C%20f%27%27%28x%29%20=%20e%5E%7B-x%20%5Cln%7Bx%7D%7D%5Cleft%20%28-%5Cln%7Bx%7D-1%20-%20%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%20%29%20=%20%5Cleft%20%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%20%29%5Ex%20%5Cleft%20%28-%5Cln%7Bx%7D-1%20-%20%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%20%29%20%5C%5C%20f%27%27%281%29%20=%20-2" title="f(x) = \left (\frac{1}{x} \right )^x =e^{-x \ln{x}} \\ f'(x) = (-\ln{x}-1)e^{-x \ln{x}} \\ f''(x) = e^{-x \ln{x}}\left (-\ln{x}-1 - \frac{1}{x} \right ) = \left (\frac{1}{x} \right )^x \left (-\ln{x}-1 - \frac{1}{x} \right ) \\ f''(1) = -2">

## Comments

## Leave a comment