dy/dx=cos(x)-6sin(2x)
The derivatives are calculated as follows:
sin(x+dx)=sin(x)cos(dx)+cos(x)sin(dx)=sin(x)+dxcos(x) because cos(dx) approaches 1 as dx approaches zero, and sin(dx) becomes dx as dx approaches zero. Therefore sin(x+dx)-sin(x)=dxcos(x); d(sin(x))/dx=cos(x).
Similarly, cos(2x+2dx)=cos(2x)cos(2dx)-sin(2x)sin(2dx)=cos(2x)-2dxsin(2x), and cos(2x+2dx)-cos(2x)=-2dxsin(2x). Therefore, d(3cos(2x))/dx=3d(cos(2x))/dx=-6sin(2x).