Given Eqn. is -1=(x+y)^2-xy-e^(3x+7y), or,
xy+e^(3x+4y)=(x+y)^2+1.
Diff. both sides,
(xy)'+{e^(3x+\color{red}{7}y)}'={(x+y)^2}'+0.
:. xy'+yx'+e^(3x+7y)(3x+7y)'=2(x+y)(x+y)'
:.xy'+y+e^(3x+7y){(3x)'+(7y)'}=2(x+y)(x'+y')
:. xy'+y+e^(3x+7y)(3+7y')=2(x+y)(1+y'), i.e.,
xy'+y+3e^(3x+4y)+7y'e^(3x+7y)=2(x+y)+2y'(x+y).
:. xy'+7y'e^(3x+7y)-2y'(x+y)=2(x+y)-y-3e^(3x+4y).
:. y'(x+7e^(3x+7y)-2x-2y)=2x+2y-y-3e^(3x+4y), or,
y'(7e^(3x+7y)-x-2y)=2x+y-3e^(3x+4y).
Hence, y'={2x+y-3e^(3x+7y)}/(7e^(3x+7y)-x-2y).
y' can further be simplified, as below :-
For this, we write the given eqn. as e^(3x+7y)=(x+y)^2+1-xy=x^2+xy+y^2+1, & submit the value of e^(3x+7y)) in y' to give,
y'={2x+y-3(x^2+xy+y^2+1)}/{4(x^2+xy+y^2+1)-x-2y}.