What allows us to perform calculus on real numbers, complex numbers etc. but prevents us from performing calculus on for example cyclic groups. what properties must exist in an algebraic structure before we can even consider taking the derivative of it?
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You might be interested in Why can't calculus be done on the rational numbers? – wgrenard Mar 22 '18 at 02:32
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You can also do calculus over the $p$-adic numbers.
And derivatives of rational functions can be defined over any field.
Robert Israel
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Well, differentiation is basically useless in the $p$-adics, but (definite) integration is very fruitful. – anon Mar 22 '18 at 02:44