The geometric integral (aka product or multiplicative integral) is defined as:
$$\prod_{a}^{b}f(x)^{\mathrm{d}x}=\lim_{\Delta x\to 0}\prod {f(x_{i})^{\Delta x}}=\exp \left(\int_{a}^{b}\ln f(x) \mathrm{d}x\right)$$
The inverse operation, the geometric derivative, is sometimes denoted with an asterisk à la Lagrange's notation, for example in the fundamental theorem of geometric calculus:
$$\prod_{a}^{b}f^{*}(x)^{\mathrm{d}x}={\frac {f(b)}{f(a)}}$$
Now, what is the Leibnizian notation for the geometric derivative? Given that Leibniz's notation for the ordinary (additive) derivative,
$$\frac{\mathrm{d}f}{\mathrm{d}x} \quad\rightarrow\quad \int_a^b \frac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x=f(b)-f(a),$$
is reminiscent of the limit definition and involves the inverse of the additive integral's product (i.e., $f(x)\mathrm{d}x$), I assumed
$$\sqrt[\mathrm{d}x\enspace]{\mathrm{d}f} \quad\rightarrow\quad \prod_{a}^{b} \left(\sqrt[\mathrm{d}x\enspace]{\mathrm{d}f}\right)^{\mathrm{d}x}={\frac {f(b)}{f(a)}}$$
which seemed like the analogous inverse of $f(x)^{\mathrm{d}x}$. Is this correct? Or acceptable in the absence of a prevailing convention?