Question

differentiate y=(Inx)^squart root x Do not simplify answer

Answer 1

\[y=(\ln x)^{\sqrt{x}}\]

Answer 2

Hey Jess :)
We can uhhh.. use logs I suppose, that should help

Answer 3

So umm.. ya ya let's try that.
Let's take the natural log of each side.\[\large\rm \ln y= \ln\left[(\ln x)^{\sqrt x}\right]\]

Answer 4

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

Answer 5

im somewhat getting it

Answer 6

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.

Answer 7

Product rule on the right, ya?

Answer 8

Too much? :O
Brain esplode?

Answer 9

no im still here

Answer 10

So then uhhh take some derivatives :)
remember derivative of \(\large\rm \sqrt{x}\) ?

Answer 11

http://openstudy.com/study#/updates/559c9c4ae4b0564dd2d41b6a