Then, where to read more about this? People who built autograd and other frameworks like Pytorch, mxnet, etc. should have learnt them in details somewhere. Where? AFAIK mxnet came out of academia (probably CMU).
Then you can go and read this fabulous blog post [2], and if you like what you see, you go to the framework built by its author, called Small Pebble [3]. Despite the name, it's not all that small. If you peruse the code you'll get some appreciation of what it takes to build a solid autodiff library, and if push comes to shove, you'll be able to build one yourself.
[1] https://www.youtube.com/watch?v=q8SA3rM6ckI
A nice overview was Backpropagation through time: what it does and how to do it, 1990. The rule itself is stated very clearly there, but without proof. The proof can be found in Maximizing long-term gas industry profits in two minutes in lotus using neural network methods, 1989 (which I believe was copied over from his earlier thesis, which I could never find a copy of).