If xsqrt(1+y)+ysqrt(1+x)=0,then dy/dx is

1.x+1/x
2.1/1+x
3.-1/(1+x)^2
4.x/1+x

given xsqrt(1+y)+ysqrt(1+x)=0
xsqrt(1+y)=-ysqrt(1+x)
squaring on both sides
x^2(1+y)=y^2(1+x)
x^2-y^2+x^2y-xy^2=0
(x-y)(x+y)+xy(x-y)=0
x=y or x+y+xy=0
x=y does not satisfy the given equation hence
x+y+xy=0
y=-x/1+x
dy/dx=-[(1+x).1-x]/(1+x)^2
=-1/(1+x)^2

Using implicit differentiation, define

F(x,y) = x*sqrt(1+y)+y*sqrt(1+x)

Then, dy/dx = -(dF/dx)/(dF/dy). The required derivatives are

dF/dx = sqrt(1+y) + y/(2*sqrt(1+x))
dF/dy = x/(2*sqrt(1+y)) + sqrt(1+x)

dy/dx = - [ sqrt(1+y) + y/(2*sqrt(1+x)) ] / [ sqrt(1+x) + x/(2*sqrt(1+y)) ]

which can be rationalized after multiplying the numerator and denominator by sqrt(1+x)*sqrt(1+y)

dy/dx= - [ (1+y)*sqrt(1+x) + (y/2)*sqrt(1+y) ] / [ (1+x)*sqrt(1+y) + (x/2)*sqrt(1+x) ]