Hi again Chipset3600!
The proper formula is:
I have some trouble understanding your scans, but they do not look right.
Let's define .
And let's define .
What is ?
Well, I couldn't make sense of your first scan, so just now I skipped to the second one.
Your partial derivatives are correct (except for that should have a minus sign).
However, for the total derivative du you shouldn't have added them as you did.
Each of them should be multiplied by dx, dy, and dz respectively.