Recently I was thinking about what a multiplicative derivative and integral would be. I came up with:

rf^{dx} = lim*h->0* (f(x+h)/f(x))^{1/h}

which is the geometric derivative. Of course after googling it I found out about the geometric and bigeometric derivatives.

Next I tried using a different operation. Analogous to the harmonic mean, I defined harmonic addition which is

a⊕b = [a^{-1} + b^{-1}]^{-1} and the harmonic difference a⊖b = a⊕(-b)=[a^{-1} – b^{-1}]^{-1} And the harmonic derivative

hf dx = = lim*h->0* h [f(x+h) ⊖ f(x)]

h is multiplied by the difference because of the natural repetition operator.

x = x

x ⊕ x = x / 2 x ⊕ x + … + x ⊕ x = x / n

Repeated addition is multiplication, so the standard derivative multiplies the function difference by 1/h. Repeated multiplication is exponentiation, so the geometric derivative takes the function ratio to the power 1/h. Repeated harmonic addition is division, so the harmonic derivative uses 1/(1/h) * a = h * a. It’s probably skipping over a bunch of details though. Obviously after doing this I found the derivatives of the usual common functions

h[x^{-2}] dx =1/(2x) and found how each was related to the normal derivative

rf^{dx} = e^{f’/f}

I haven’t been able to find anything talking about this harmonic derivative. I found sources talking about infinite families of Non-Newtonian calculi but they only ever talked about geometric and bigeometric. One site did mention quadratic and harmonic derivatives, but only in passing. Also there were a lot of papers that seemed beyond my level (calc 1 and 2, lin algebra, and general self study)

Basically I’m looking for the correct terminology and resources to look into this more. As well I’m wondering if there are any other notable generalized derivatives. Perhaps some that aren’t just

EDIT: Sorry if the formatting is terrible. I composed this on mobile during lecture lol

Tried fixing the formatting. I may have forgotten an negative sign. I’m trying to figure out where specifically I got my formula for it from. It doesn’t change the idea of the post though.

The article mentioning the quad and harm. http://planetmath.org/nonnewtoniancalculus

I think I didn’t give this article a fair chance at first. It’s seems to answer my questions. https://www.tandfonline.com/doi/pdf/10.1080/0020739790100406