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I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0):

F(x+Δx) - F(x) = F'(x) * Δx

The Δx factor ensures that the RHS remains 0 when the LHS is 0.

And we get the familiar

(F(x+Δx) - F(x)) / Δx = F'(x)

I'd like to think of it as taking infinitesimal differences over some infinitesimal interval, and I wondered what would it look like if I were to take infinitesimal ratios instead. And it came out looking like this:

F(x+Δx) / F(x) = F*(x) ^ Δx

The Δx exponent ensures that the RHS remains 1 when the LHS is 1.

Then it comes out as

(F(x+Δx) / F(x)) ^ (1/Δx) = F*(x)

I've tried to use this on one sample function, F(x) = x^2; the results are F'(x) = 2x and F*(x)=e^(2/x)

If F'(x) means the rate of change of between two things (like velocity when F(time) is position), then what does F*(x) mean?