Question

use log differentiation for x^ln x

Answer 1

\[\large y=x^{\ln x}\]Step 1 is to take the natural log of both sides.\[\huge \ln y=\ln(x^{\ln x})\]From here, we want to apply a rule of logarithms,\[\large \log(a^b)=b\cdot \log(a)\]Applying this to our problem gives us,\[\large \ln y=\ln(x^{(\ln x)}) \qquad \rightarrow \qquad \ln y=(\ln x) \ln(x)\]

Answer 2

The whole goal was to try and get the variable X out of the EXPONENT position, because it's difficult to differentiate when we have a variable up there.

We have successfully done that, the (ln x) is no longer an exponent. So from here we can take the derivative, giving us,\[\large \ln y=(\ln x)^2 \qquad \rightarrow \qquad \frac{1}{y}y'=2(\ln x)(\ln x)'\]Understand how I got this derivative?
The prime on the second log term there is due to the chain rule, we still need to differentiate that part.