What is $dfrac{partial x}{partial t}$ in a progressive wave?

In the general sense, it is wrong to think about the derivative (partial or total) $frac{partial f(x)}{partial x}$ as a fraction. For a finite difference, it is fine, but since the derivative is the infinitesimal limit of a fraction, it is not necessarily well-defined how to treat numerator and denominator separately.

Rather, for full derivatives you might use the chain rule $frac{d f}{d x} = frac{d f}{d t} frac{dt}{dx} $, which, however, is more complicated in the case of partial derivatives. The most straightforward way of finding $frac{partial x}{partial t}$ might be to invert $y(x)$ to find $x(y)$ and then taking the time derivative directly.

Note that in physics we often perform substitution of variables in integrals by pretending the fraction is well-defined. One must be very careful, since this only works under certain circumstances.