To be honest, I never get what people want with all that business and wonder if it is because the abstraction ("ordered derivatives") implied is not ideal.

If we follow the ordinary chain rule (for a single coordinate if you want) through the edges of the computational (DAG) graph, we get the right thing in each step.

The only other rule you need is that "if you use one variable several times in a calculation (i.e. several edges from(fw)/to(bw) the same node), you need to add the gradients computed for each", but IMHO that is pretty basic and intuitive, too. (So if you plug in z for both x and y into f(x, y), you have d/dz f(z, z) = f_x(z, z) + f_y(z, z), where the subscript indicates partial derivative.)

To me this seems both mathematically simpler than mixing the two into a "more than chain rule" thing and closer to what is actually going on algorithmically in a given implementation (the one I'm most familiar with is probably PyTorch's).