Simple Harmonic Motion Question

Hi. My Physics teacher never explains anything, and there's something I'm very confused about. I need to know before school starts again, so I'm asking here.
***I'm using w for the small letter omega.***

At a random position in SHM, displacement is given as
x=Acos(wt), where A is the amplitude.

To find the equation for velocity at that point, you just derive it. The derivation of cos(x) is -sin(x). All is fine so far.

What I'm confused about is why the equation is x= -wAsin(wt), and not just
x= -Asin(wt). Where does the extra omega come from??

I've noticed that when you derive it again, you get w^2, so clearly it is multiplied by w each time, but I can't work out why.

I've asked a few people and nobody could explain it to me. No one is my class seems to have any idea either.

Any and all help is greatly appreciated. Thanks!

When you differentiate Acos(wt) you use the chain rule, which basically meant you multiply the derivative of cos(u) by the derivative of u, where u = wt here. Therefore you end up with -Asin(wt) x w = -Awsin(wt).