anonymous
  • anonymous
differentiate y=(Inx)^squart root x Do not simplify answer
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
\[y=(\ln x)^{\sqrt{x}}\]
zepdrix
  • zepdrix
Hey Jess :) We can uhhh.. use logs I suppose, that should help
zepdrix
  • zepdrix
So umm.. ya ya let's try that. Let's take the natural log of each side.\[\large\rm \ln y= \ln\left[(\ln x)^{\sqrt x}\right]\]

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zepdrix
  • zepdrix
Now we can apply our log rule to rewrite the right side,\[\large\rm \ln y=\sqrt{x} ~\ln\left[(\ln x)\right]\]And from there you'll apply product rule :) It's gonna get messy! Confused by any of that? :o
anonymous
  • anonymous
im somewhat getting it
zepdrix
  • zepdrix
So let's first `set up` our derivative so we understand how it's going to look.\[\large\rm \color{royalblue}{(\ln y)'}=\color{royalblue}{(\sqrt x)'}\ln(\ln x)+\sqrt x\color{royalblue}{\left[\ln(\ln x)\right]'}\]The blue stuff is what we need to differentiate.
zepdrix
  • zepdrix
Product rule on the right, ya?
zepdrix
  • zepdrix
Too much? :O Brain esplode?
anonymous
  • anonymous
no im still here
zepdrix
  • zepdrix
So then uhhh take some derivatives :) remember derivative of \(\large\rm \sqrt{x}\) ?

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