To avoid Quotient Rule for Diffn., we first, rewrite the given eqn. as,
#xe^y=(2y-x)(y-4)=2y^2-xy-8y+4x, or,#
#xe^y+xy+8y=2y^2+4x=2(y^2+2x).#
#:. d/dx(xe^y+xy+8y)=d/dx{2(y^2+2x)}=2d/dx(y^2+2x).#
Following the Usual Rules of Diffn., we get,
#:.d/dx(xe^y)+d/dx(xy)+8d/dx(y)=2{d/dx(y^2)+2d/dx(x)}.#
Here, in the L.H.S., by the Product Rule, we have,
# d/dx(xe^y)=xd/dx(e^y)+e^yd/dx(x),#
#=xe^yd/dx(y)+e^y..........[because," the Chain Rule],"#
#:. d/dx(xe^y)=xe^ydy/dx+e^y.#
Also, #d/dx(xy)=xd/dx(y)+yd/dx(x)=xdy/dx+y.#
In the R.H.S., we have, #d/dx(y^2)=2yd/dx(y)=2ydy/dx.#
Altogether, we get,
# xe^ydy/dx+e^y+xdy/dx+y+8dy/dx=2{2ydy/dx+2}=4ydy/dx+4#
#:. (xe^y+x+8-4y)dy/dx=4-e^y-y,#
#rArr dy/dx=(4-e^y-y)/(xe^y+x+8-4y).#
Enloy Maths.!
