Derivation of the speed of sound waves equation

[FerPhys]

I'm learning about the speed of sound waves through a medium. The derivation is initiated through Impulse=change in linear momentum (I=Δp), then I=ΣFΔt=(Area×Δpressure×Δt) in the x direction
The derivation proceeds by replacing the Δpressure with another equation we had derived earlier
ΔP=-B(ΔV/Vi) where B is the bulk modulus. The initial volume (Vi) is vAΔt where v is the velocity of sound multiplied by the time it takes to reach the end of the container (gives you a length which you can multiply to get Area to get the volume of the gas without anything force acting on it).
This is where I get a bit confused. My textbook says ΔV= (-v_xAΔt) where v_x = the speed of the elements in the medium or in this case a gas. When you plug everything back in into -B(ΔV/Vi) you get ΔP=B(v_x / v) . My question now is, why did the Δt cancel out? Wouldn't that be like saying that the time it took for those gas particles to move through the entire container was the same as the time it took for sound to move through the entire container? Also, if that were the case wouldn't they be going at the same speed and therefore the vx/v should also cancel?

This my first post on this website so if I can do anything to make my posts more clear please let me know!
Thank you!

 

[drvrm]

Are you deriving the Newton's formula for speed of sound in say air/gas?

 

[nasu]

Can you show an image of the page where they do this?

 

[FerPhys]

This may come off as completely rude but I understand the derivation now.. sorry :(
yes its for Newtons formula for speed in sound..