(Hyper) Dual Numbers for automatic differentiation (CAS)

[mcasl]

Hi! Thank you for your interest.The context is that I'm teaching automatic differentiation and I propose my students simple exercises like the one I showed for them to solve using pen and paper. Thus the convenience of having a tool to assess the results.
I also tried the matrix approach, and I was in awe watching how xcas was capable of handling to perfection trigonometric expressions. In fact, it caused me great curiosity how xcas was smart enough to compute them.
I can think of solutions to the topic involving loops and substitution, or using symb2poly and further string processing, but I was intrigued in knowing if xcas was capable of learning the rules for the \( \epsilon_i \) and apply them in its intermediate computations, not just relying on user at the end of the process for discarding terms.