The title pretty much says it all. I'm trying to find a linear transform, maybe a vague analog of a derivative, that has the property that if $f(x)=ab^x$, then $T(f)=ab(b-1)^x$, analogous to the property of the derivative that if $f(x)=x^n$, then $f'(x)=nx^{n-1}$. I can't figure out how to find an explicit form that can be applied to other functions.
I've tried proving the aforementioned property of derivatives and working my way backwards, but haven't gotten anywhere. It seems to me that there should be infinite such transforms--I only need a single example, but more variety is always better--but I don't really know very much about the area, so for all I know it's impossible.
Additional: this isn't really part of the main question, but if this transform does exist, would it have a graphical representation as the derivative does, and if so what?
No such transformation exists, provided that $b$ is an arbitrary constant:
On one hand, $T(2^{x + 1}) = 2(1)^{x + 1} = 2$.
On the other, $T(2^{x + 1}) = T(2 \cdot 2^x) = 2 T(2^x) = 2(2(1)^x) = 4$, a contradiction.