Complicated derivative question

[sooyong94]

Homework Statement

How to I find the derivative of
$y=\sqrt{\sqrt{(1+\sqrt{x})}-1}$

Homework Equations

Logarithmic derivative

The Attempt at a Solution

I decided to take natural logs to both sides, and this is what I have:
$\ln y=\frac{1}{2} \ln (\sqrt{(1+\sqrt{x})}-\frac{1}{2}$
$\ln y=\frac{1}{4} \ln(1+\sqrt{x})-\frac{1}{2}$
Then an implicit derivative gave me this:
$\frac{1}{y} \frac{dy}{dx}=\frac{1}{8} (\frac{1}{\sqrt{x}+x})$

Now when I multiply both sides by y, the expression becomes very complicated. How do I manipulate them? :(

 

[maajdl]

Why would you want to simplify the expression?
Just leave it as it is.

 

[sooyong94]

The reason is I need to find dy/dx in terms of x. :(

 

[maajdl]

Then substitute the expression for y, and that's it.

 

[Devils]

http://en.wikipedia.org/wiki/Chain_rule

 

[sooyong94]

Chain rule seems complicated... :(

 

[DrClaude]

I decided to take natural logs to both sides, and this is what I have:
$\ln y=\frac{1}{2} \ln (\sqrt{(1+\sqrt{x})}-\frac{1}{2}$

You made a mistake there.
$\ln y=\frac{1}{2} \ln ( \sqrt{(1+\sqrt{x})}-1)$

 

[sooyong94]

Wasn't it looks the same when expanded?

 

[DrClaude]

Wasn't it looks the same when expanded?

When expanding what? $\ln(a+b) \neq \ln(a) + \ln(b)$

 

[sooyong94]

Oops, my bad. :P