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Multiplicative Infinitesimals
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For the second, let's start by expanding (quasi-)Leibniz notation for the derivation into the conventional limit definition. The corresponding definitions in multiplicative calculus sometimes use this subtraction, but also use quotients instead of differences of expressions involving the limit variable and where its going. $${\huge \mathscr{P}} \left( \sqrt[d x] {q f(x)} \right)^{d x}$$$${\huge \mathscr{P}} q f(x)$$$$f$$I checked out Nonstandard analysis and Hyperreal number on Wikipedia, which I had been meaning to do for a while, to see whether these multiplicative infinitesimals are already "handled" by it.
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