I actually divided the velocity of a particle in a progressive wave $\dfrac{\partial y}{\partial t}$ to $\dfrac{\partial y}{\partial x}$ and got $\dfrac{\partial x}{\partial t}$. Which is equal to $\dfrac{-\omega}{k}$ . Mathematically it is equal to $-V$. But I am not getting its qualitative meaning.
I divided the two highlighted equations Also, can I even divide two derivatives which have different constants$?$
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.
