Use logarithmic differentiation to calculate the derivative:
f(x) = (1+sin(x))^(1/x)

Question

Use logarithmic differentiation to calculate the derivative:
f(x) = (1+sin(x))^(1/x)

anonymous · Answer

I know you take the ln of both sides, but I get stuck because I don't know what to do with the power!! Help? Pretty please?

hartnn · Answer

$\log x^n = n \log x$
$\log (1+\sin x)^{1/x} = (1/x) \log (1+\sin x)$

anonymous · Answer

Okay, I got that far, but I'm stuck!

anonymous · Answer

...unless you use the product rule, which I think you don't do...

hartnn · Answer

you have to use product rule, and ofcourse chain rule for log(1+sin x)

anonymous · Answer

Okay, just a sec while I work this out then...

hartnn · Answer

sure, take your time and tell me what u get ?

anonymous · Answer

Final answer: 
$$(1+\sin(x))^{1/x}*(\frac{ \cos(x)}{(x)(1+\sin(x)} -\frac{ \ln(\sin(x) }{ x ^{2} })$$ 
I'm giving the equation doodad down there a whirl, forgive any errors! I got the answer in my book, too, so it's right! Thanks for all the help, we literally just learned this yesterday!

hartnn · Answer

welcome ^_^
you learned fast! good :)

anonymous · Answer

Thanks, I pick things up pretty quick! And I love your avatar/icon!

hartnn · Answer

me too :D