Math Help - Logarithmic differentiation

1. Logarithmic differentiation

Can someone tell me if this is correct?
y = (lnx)^x
y' = x(lnx)^x-1 * 1/x
y' = (lnx) ^ x-1

It seemed a bit too simple so I wanted to check if I'm doing it right, thanks.

2. Re: Logarithmic differentiation

It's not correct.
Write $y = [\ln(x)]^x = e^{x\ln[\ln(x)]}$
Now, compute $\frac{dy}{dx}$

3. Re: Logarithmic differentiation

I'm trying to understand what you did there, but I also have a second method,
Which is: lny = xln(lnx)
Would this also be correct?
So...
y'/y = ln(lnx) + x * 1/lnx * 1/x
y' = y(ln(lnx) + 1/lnx)
y' = (lnx)^x(ln(lnx) + 1/lnx)
Or is this also wrong?

4. Re: Logarithmic differentiation

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

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

$\ln{y} = x \cdot \ln(\ln{x})$

$\frac{y'}{y} = x \cdot \frac{1}{x\ln{x}} + \ln(\ln{x}) \cdot 1$

$\frac{y'}{y} = \frac{1}{\ln{x}} + \ln(\ln{x})$

$y' = y\left[\frac{1}{\ln{x}} + \ln(\ln{x})\right]$

$y' = (\ln{x})^x\left[\frac{1}{\ln{x}} + \ln(\ln{x})\right]$