I had followed this when I was learning DL through Andrew NG's course. In one of the lessons, he had the formula for calculating the loss as well as it's derivatives.

I tried driving these formulas from scratch using what I learned from OP's post but it felt like there was something missing. I think it boils down to me not knowing how to aggregate those element wise derivatives into a matrix form. Afaik the Matrix cookbook and certain notes from Stanford cs231n that helped me grok it fully