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i mean like a specific formalism (ideally one that can fade into the background once learned) the dual numbers are a good example, you just make up a number e such that e * e = 0 then you can differentiate f(x) by just evaluating (f(x+e)-f(x))/e (fun thing to try out)
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another "system" would be non-standard analysis this is where you pretend there's a number H that's like infinity but still a natural number, and then you can use 1/H as your infinitesimal this one makes integrals nice, since you're just adding up H slices of width 1/H
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Sounds like you’d need at least 2 consistent properties to make a system work for both differentiation and integration… why would you expect this to exist? At least with e*e=1 or i^2=-1 you’re just making one thing up
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