Look at each term and find its derivative.
xyeˣ: d(xy)/dx=xdy/dx+y; d(eˣ)/dx=eˣ; d((xy)eˣ)=xyeˣ+eˣ(xdy/dx+y). When x=0, this is y.
ye⁻ˣˠ: d(e⁻ˣˠ)/dx=e⁻ˣˠ(-xdy/dx-y); d(y(e⁻ˣˠ))=y(e⁻ˣˠ(-xdy/dx-y))+e⁻ˣˠdy/dx. When x=0, this is -y²+dy/dx.
sin²(x): d(sin²(x))=2sin(x)cos(x). When x=0, this is 0.
Put the three derivatives together: y-y²+dy/dx=0, so dy/dx=y²-y.