Why is the speed of a quantum particle defined as coefficient of $t$ over coefficient of $x$?

The following may be a useful approach.

A simple traveling wave can be written as:

$$ y=sinleft(kx-omega t right) $$

We want to follow the position of the wave at a constant phase, $phi$. Let that phase=0 which leads to:

$$ phi=kx - omega t = 0 $$

$$ kx=omega t $$

$$ x=frac{omega}{k} t $$

Then the velocity will be

$$ v=dot{x}=frac{omega}{k} $$

which is the ratio of the coefficient of $t$ over the coefficient of $x$.

I hope this helps.