Computing Partial Derivative for RTRL training for Recurrent Neural networks

Computing Partial Derivative for RTRL training for Recurrent Neural networks

Ok so first off Hi,
Secondly its 3am and I promise you I am fried after spending 3 solid days
I understand RNN's and RTRL, but my calculus brain has ran away from me at
this point in time.
Basically I'm at the stage where I Need to calculate this:

I used a variety of sites and my textbook (which had 0 on this subject),
but this is the rest of my primary source willamette.edu
The issue I'm having is how to programmatically (Java) partially
differentiate
Yk in the direction Wij
I can't wrap my head around how to go about that.
NOTE: I do actually understand how the RNN and RTRL works, confidently.