The Julia ecosystem provides has a library that includes the differentiation rules hinted at at the end.

https://github.com/JuliaDiff/ChainRules.jl is used by (almost all) automatic differentiation engines and provides an extensive list of such rules.

If the example used sin|cos the auto diff implementations in Julia would have called native cos|-sin and not encurred such a "truncation error". However the post illustrates the idea in a good way.

Good post oxinabox