We can simplify the given equation:
x√(1+y)=-y√(1+x), x²(1+y)=y²(1+x), x²+x²y-y²-xy²=0.
This factorises: x²-y²+x²y-xy²=(x-y)(x+y)+xy(x-y)=0=(x-y)(x+xy+y).
But x=y is not a solution of the original equation so we can remove the factor x-y:
x+xy+y=0 so y(x+1)=-x, y=-x/(x+1) which can be differentiated wrt x:
dy/dx=(-(x+1)+x)/(1+x)²=(-x-1+x)/(1+x)²=-1/(1+x)².