Derivatives using Logarithmic Differentiation

[howsockgothap]

Homework Statement

Using logarithmic differentiation calculate the derivative of y=e^(x^x)

The Attempt at a Solution

y=e^x^x
LNy=LNe^x^x
LNy=x^xLNe
...
Stuck!

This seems to be the only way you can do it, but once I get to that part I'm not sure what else there is to do. I know lne=1, so does that mean the answer is y'=e^(x^x)*x^x

 

[arildno]

$$\ln(\ln(y))=\ln(x^{x}\ln(e))=\ln(x^{x})=x\ln(x)$$

And:
$$(\ln(\ln(y))'=\frac{y'}{y\ln(y)}=\ln(x)+1$$
whereby follows:
$$y'=y\ln(y)(\ln(x)+1)=e^{x^{x}}x^{x}(\ln(x)+1)$$

 

[micromass]

Maybe you need to take the logarithm two times...

 

[howsockgothap]

I don't see how it goes from xLNx to LNx+1