# Complicated derivative question

1. Feb 18, 2014

### sooyong94

1. The problem statement, all variables and given/known data

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

2. Relevant equations
Logarithmic derivative

3. 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? :(

2. Feb 18, 2014

### maajdl

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

3. Feb 18, 2014

### sooyong94

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

4. Feb 18, 2014

### maajdl

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

5. Feb 18, 2014

### Devils

6. Feb 18, 2014

### sooyong94

Chain rule seems complicated... :(

7. Feb 18, 2014

### Staff: Mentor

$\ln y=\frac{1}{2} \ln ( \sqrt{(1+\sqrt{x})}-1)$

8. Feb 18, 2014

### sooyong94

Wasn't it looks the same when expanded?

9. Feb 18, 2014

### Staff: Mentor

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

10. Feb 18, 2014