#csc (x^2+y^2)=e^(-x)-y#
differentiate both sides of the equation with respect to x
#d/dx(csc (x^2+y^2))=d/dx(e^(-x)-y)#
#-csc (x^2+y^2)*cot (x^2+y^2)*d/dx(x^2+y^2)=d/dx(e^(-x))-d/dx(y)#
#-csc (x^2+y^2)*cot (x^2+y^2)(2x^2+2y*y')=(e^(-x))*(-1)-y'#
#-csc (x^2+y^2)*cot (x^2+y^2)(2x^2+2y*y')=-e^(-x)-y'#
Multiplication then transposition
#(2x^2(-csc (x^2+y^2)*cot (x^2+y^2))+e^(-x)=#
#2y*y'*(csc (x^2+y^2)*cot (x^2+y^2))-y'#
factor out the y'
#-2x^2csc (x^2+y^2)*cot (x^2+y^2)+e^(-x)=#
#[2y*csc (x^2+y^2)*cot (x^2+y^2)-1]y'#
Divide both sides by #[2y*csc (x^2+y^2)*cot (x^2+y^2)-1]#
#y'=(-2x^2*csc (x^2+y^2)*cot (x^2+y^2)+e^(-x))/[2y*csc (x^2+y^2)*cot (x^2+y^2)-1]#
God bless....I hope the explanation is useful