Yesterday's smbc had an interesting mathematics challenge:
https://www.smbc-comics.com/comic/derivative
A = The number of symbols in a function
B = The number of symbols in that function's derivative.
Homework: Find the function of finite length with the highest ratio of B to A.
Now, obviously you have to decide for yourself what 'symbols' means, but after doing that, what's the best you can come up with?
I asked a friend who came up with a very pleasing solution, but it might be possible to do even better:
https://xeno.chat/@bj/108199208965188517
D(e^x)=e^x
D(e^(e^x))=e^(e^x) e^x, thanks to the chain rule
D(e^(e^(e^x)))=e^(e^(e^x)) e^(e^x) e^x, and so on
The derivative of an exponent-tower of e^e^e^...^x is a product of these towers starting from the original and going down to e^x. The number of symbols on the left side increases linearly, but the number of symbols on the right increases quadratically, so the higher you make your tower, the higher your ratio is, and you can make the ratio arbitrarily large.
Obviously if you accept this answer,
the question evolves past 'what is the highest ratio' into 'what's the fastest growing function you can write to produce higher ratios faster? Can you make bigger polynomials? Exponentials? Bigger?'