Let us first bring up inverse function theorem : If $y=f(x)$ and if $f'(x)$ exists at some point $x=a$, then exists in some neighborhood of $(a,f(a))$ an inverse function $f^{-1}(x)$ which around corresponding $b = f(a)$, such that it is differentiable and that this differential fulfills:
$$(f^{-1})'(b) = \frac{1}{f'(a)}$$
Could this bring a fruitful approach to define the reciprocal infinitesimal quotient below:
$$\frac{\partial f^{-1}(y)}{\partial y}(b) = \frac{\partial x}{\partial f(x)}(a)$$
Would it lead to anything meaningful, or will we run into trouble if we would try to do so?
For context : My mind wanders to some example $$\frac{\partial \sin(t)}{\partial \cos(t)}$$ which in polar coordinates $x = \cos(t), y = \sin(t)$ and trigonometric identity could be intuitively interpreted as the differential of the function describing the upper part of unit circle:
$$\frac{\partial \sqrt{1-x^2}}{\partial x}$$
Some far fetched but interesting try would be to extend to find meaning and interpretation to things like $$\frac{\partial g(t)}{\partial h(t)}$$
If we stick with $\mathrm d$ rather than $\partial$, then there's little problem with handling single-variable in this sort of way. We define $\mathrm d (f(t))=f'(t)\mathrm dt$ (and similarly for any other variable), and then the algebra works out nicely for first-order derivatives.
For example: $$\dfrac{\mathrm{d}\sin t}{\mathrm{d}\cos t}=\dfrac{\cos t\,\mathrm{d}t}{-\sin t\,\mathrm{d}t}=-\cot t\text{.}$$ And we also have, if $t\in[0,\pi]$ and we let $x=\cos t$: $$\dfrac{\mathrm{d}\sin t}{\mathrm{d}\cos t}=\dfrac{\mathrm{d}\sqrt{1-x^{2}}}{\mathrm{d}x}=\dfrac{-2x\,\mathrm dx}{2\sqrt{1-x^{2}}\,\mathrm dx}=\dfrac{-x}{\sqrt{1-x^{2}}}=\dfrac{-\cos t}{\sin t}=-\cot t\text{.}$$
Now, there could be trouble with second-order derivatives, but there is also a fix for this (basically, don't write a second derivative as $\dfrac{\mathrm d^2y}{\mathrm dx\vphantom{)}^2}$ if $x$ and $y$ could depend on some other variable $t$). Details are discussed in the recently-popular-on-the-internet paper "Extending the Algebraic Manipulability of Differentials" by Bartlett and Khurshudyan. One version can be found on the arXiv at here and another is on the site for the journal "Dynamics of Continuous, Discrete and Impulsive Systems" here.