Complex derivative of x: ((x)^(1/x))'

[Ondina]

<<Moderator note: Remember that filling in the complete homework template is mandatory in the homework forums. This thread has not been deleted due to containing relevant replies.>>

1. Homework Statement
((x)^(1/x))'

Homework Equations

This probably isn't overly dificult, but it has got me stumped, if anyone would be so kind as to reply, would you please post the entire process, so that I can try to better understand it. Thank you very much

The Attempt at a Solution

 

[BvU]

Hello Ondina, welcome to PF !

<<Moderator note: Removed comment on now deleted full solution.>>

That way you are robbed from a learning experience that might have helped you in a future exercise.

If you see something like $x^{1\over x}$ it's sometimes helpful to write it as $e^{\ln x\over x}$. To take the derivative you use the chain rule: $$\left (e^{\ln x\over x} \right )'= e^{\ln x\over x} \; \left ({\ln x\over x} \right )'
$$.

 

[Chestermiller]

BvU is making use of the relationship that:

$y = e^{\ln y}$

This follows from the definition of the natural log of a quantity: lny is the power you have to raise e to in order to get y.

Chet

 

[Ondina]

Thank you all very much for your help!