could 𝑓’⁻¹ technically be used as integral notation

[wary forge]

If I'm understanding this question correctly, you're asking the difference between the inverse of a function I.E.:

𝑓’(x)
vs:
𝑓⁻¹(x)

The first is the derivative of the function 𝑓(x) with respect to x which is an analytical method of finding the slope at any point x. The second would be 1/𝑓(x) or the inverse of 𝑓(x). The two of those are likely not equal. There may be a function using the natural log where they're equal, but I can't come up with one off the top of my head.. anyway, I'm not sure if that answered your question or made things less clear.

[stoic sinew]

i wouldn't write it... closest thing i can think of that is still somewhat legible is $\left(\frac{d}{dx}\right)^{-1} f(x)$

[vague meteor]

artemetra