I'm trying to make the switch from backend development to machine learning right now, and even though I have a supportive employer I've begun thinking about going back to school because the math involved seems steeper and harder to avoid.
Disclaimer: My math level is high school since I self-taught my way into software engineering.
You might be interested in the machine learning library I wrote for spaCy: https://github.com/explosion/thinc . The backward propagation can be understood quite simply with callbacks.
The main problem I had, and I think you're having, is that the notation around calculus is just hopeless. It's even worse when you're trying to map between the equations and backpropagation, because what you need is slightly different. I also find the notations for functions really bad.
The other thing you need to get used to are the ways linear algebra overload the operators. This is just data, not concepts. Personally I find it much easier to just work in the Einstein summation: https://obilaniu6266h16.wordpress.com/2016/02/04/einstein-su... . Once it's debugged, you can translate into faster calls to numpy.tensordot.
Don't go back to school. School's fucked. It'll take you three years to do about 200 hours of learning. Hiring a tutor would be vastly better if you need some help.
I can study on my own, and have been for stats/probability. But started asking myself, why not just get a degree and make it quicker. I'd have to leave my job for a number of years, but then I'd be able to study all day instead of nights/weekends.