Is there any intuitive interpretation or simplification of $\exp\left(\frac{d}{dx} \ln(f(x))\right)$?
Forms like $\phi=\exp\left(\frac{d}{dN} \ln(f(N))\right)$ are common in thermal physics/chemistry. Typically $N\gg 0$, and $f(N)$ is the ratio of two increasing, positive-definite, "very large" functions (multiplicities).
For example, $\phi(T,P,N)=\exp\left( \frac{∂}{∂N} \left( \ln Ω_{IG}(T,P,N) - \ln Ω(T,P,N) \right) \right)$ is the fugacity coefficient.
Yes, it is the multiplicative derivative, defined as
$$f^*(x):=\lim_{h\rightarrow 0}\left(\frac{f(x+h)}{f(x)}\right)^{1/h},\quad f(x)\neq0.$$
Expressed in terms of the ordinary (additive) derivative, you can easily show
$$f^*(x)=\exp \left(\frac{f'(x)}{f(x)}\right).$$
Just as the additive derivative represents "instantaneous slope," the multiplicative derivative represents "instantaneous growth rate." Linear functions have constant additive derivative, while exponential functions have constant multiplicative derivative.
