How Do I Solve This? (d^2 z)/(dx dy) = -(x ln(x))^{-1}, w/ x=y=z, ...

[Abhishekdas]

I am getting the answer as 0. Here is what I've tried

 

[tkhunny]

View attachment 10150
I am getting the answer as 0. Here is what I've tried
View attachment 10151

What is \(\displaystyle \dfrac{d}{dx}x^{x}\)?

Otherwise, I suppose I could be a fun problem to work out with the definition of a derivative.