Let y=sin(x), so dy/dx=cos(x); and let f(x)=sin3(x)=y3.
df/dx=(df/dy)(dy/dx)=3y2cos(x)=3sin2(x)cos(x).
sin(2x)=2sin(x)cos(x); sin(x)sin(2x)=2sin2(x)cos(x); sin2(x)cos(x)=½sin(x)sin(2x).
The derivative can be written df/dx=(3/2)sin(x)sin(2x).
sin2(2x)=(sin(2x))2=4sin2(x)cos2(x) is not the derivative.