I have a discovery that starts to legitimize logarithmic semi-operator calculus.

given:

q:d/dx f(x) = lim h->S(-q) [f(x{-q}h) }-q{ f(x)] }1-q{ h

I can prove that

q:d/dx e^x = e^x

or that if we lower the operators involved in the difference quotient by any value 0 <= q < 1; and change h's limit from zero (the additive identity) to S(-q) (the lowered additive identity), e^x is still it's own output.

given:

q:d/dx f(x) = lim h->S(-q) [f(x{-q}h) }-q{ f(x)] }1-q{ h

I can prove that

q:d/dx e^x = e^x

or that if we lower the operators involved in the difference quotient by any value 0 <= q < 1; and change h's limit from zero (the additive identity) to S(-q) (the lowered additive identity), e^x is still it's own output.