Given 2 functions $f(x),g(x)$ we have: $$D(f(x)+g(x)) = D(f(x))+D(g(x))$$ And $$D^*(f(x)g(x)) = D^*(f(x))D^*(g(x))$$ where $D^*$ is the multiplicative derivative: $$D^*(f(x)) = lim_{h\rightarrow 0} {\frac{f(x+h)}{f(x)}}^{1/h}$$
My question is whether there is another type of derivative $H$ such that $$H(f(x)^{g(x)}) = H(f(x))^{H(g(x))}$$