What's the math for real world back-propagation?
There is Matrix Calculus, (and I would recommend the very useful Matrix Cookbook as a bookmark to keep), but for the most part, when it comes to derivatives, it just boils down to pointwise differentiation and keeping your dimensionalities in check.

You might also want to look up Autodifferentiation. This is sort of a generalisation of the Chain Rule, such that it's possible to decompose any composite function, i.e. $a(x) = f(g(x))$, and calculate the gradient of the loss with respect to $g$ as a function of the gradient of the loss with respect to $f$.

This means that for every operation in your neural network, you can give it the gradient of the operation that "consumes" it, and it'll calculate its own gradient and propagate the error backwards (hence back-propagation)