cant understand this transformation

i know that each derivative pops iw

and

$\displaystyle \hat{y}' ->-ixy(x)$

$\displaystyle \hat{y}'(\omega) ->-ixy(x)$

x is a signs of derivative

but i dont know how its been done in here

$\displaystyle -ixy'(x)=(i\omega \hat{y}(w))'$

how they decided that is the derivative of this whole expression

muliplying by x means derivative

but here it something else

$\displaystyle

f[xy'(\omega )]=i\frac{\mathrm{d} }{\mathrm{d} \omega}f[y'(x)]=i(i\omega \hat{y}(\omega))'=i(i \hat{y}(\omega)+i\omega \hat{y}'(\omega))$

i cant see what laws they follow here

?