#y=y(x-y)^2+e^(xy) rArr d/dx(y)=d/dx{y(x-y)^2+e^(xy)}.#

#:. dy/dx=d/dx{y(x-y)^2}+d/dx{e^(xy)}.....(star).#

To differentiate the R.H.S., we use the **Product Rule** and the

**Chain Rule** :

#d/dx{y(x-y)^2}=yd/dx(x-y)^2+(x-y)^2d/dx(y),#

#=y{2(x-y)}d/dx(x-y)+(x-y)^2dy/dx,#

#=2y(x-y){d/dx(x)-d/dx(y)}+(x-y)^2dy/dx,#

#=2y(x-y){1-dy/dx}+(x-y)^2dy/dx,#

#=2y(x-y)+(x-y)^2dy/dx-2y(x-y)dy/dx,#

#=2y(x-y)+(x-y){(x-y)-2y}dy/dx,#

#=2y(x-y)+(x-y)(x-3y)dy/dx.............(1).#

Similarly, #d/dx{e^(xy)}=e^(xy)d/dx{xy},#

#=e^(xy){xd/dx(y)+yd/dx(x)},#

#=e^(xy){xdy/dx+y}=xe^(xy)dy/dx+ye^(xy).............(2).#

Altogether, by #(star), (1) and (2),#

#dy/dx=2y(x-y)+(x-y)(x-3y)dy/dx+xe^(xy)dy/dx+ye^(xy)#

# :. {1-(x-y)(x-3y)-xe^(xy)}dy/dx=2y(x-y)+ye^(xy)#

#rArr dy/dx=[y{e^(xy)+2(x-y)}]/{1-(x-y)(x-3y)-xe^(xy)}.#

**Enjoy Maths.!**