Hey everyone, I made a program to calculate derivatives symbolically. You can try it online here:
https://joshuaji.com/projects/derivative/.
I used Mathquill to display and input latex. If you have used Desmos, it is the same library they use.
It’s not perfect but it was a really fun project to make! I was inspired by this Haskell blog post:
http://5outh.blogspot.com/2013/05/symbolic-calculus-in-haskell.html.
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The Janiczek/transform might interest you - we use it in elm-in-elm for optimizing AST trees 
Sounds like something for your simplify function!
It goes from inside out, trying out various simplifications until it can’t anymore, then goes up.
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