As nobody else seems to be helping you yet you might like to know that I've actually tried all the calculations myself now and I get what you get, and at least the dq2 by dr value matches what you get when you try by the method I suggested in your first set of posts.

I've also convinced myself your method is OK by trying it on various other functions.

There's still a misprint - you've got the sign wrong for dq2/dy in your initial list of partial derivatives though it is correct in your matrix.

So I think either the solution in your book is wrong, or possibly that you have misread the problem somehow or other - perhaps by getting one of the functions wrong somehow. Let's hope someone else can shed some light on the problem.