Extremely hard for me at least!

This problem is just something I came up with. for practicing properties, and how to use partial.

I just want to know if I did this right.. Sorry, but no one I know can do this stuff, not even the cal teacher at my higschool .

consider:

Find and

Then, say

Find

I found (hopefully correctly) that

Then

Is that right?

I can't figure out the partial with respect to y though.. Can someone help?