`"If "xsqrt(1+y)+ysqrt(1+x)=0," prove that "(dy)/(dx)=-(1)/((x+1)^(2)).`

Explanation

`xsqrt(1+y)+ysqrt(1+x)=0`
`impliesxsqrt(1+y)= -ysqrt(1+x)`
`impliesx^(2)(1+y)=y^(2)(1+x)`
`impliesx^(2)+x^(2)y=y^(2)+xy^(2)`
`impliesx^(2)-y^(2)+x^(2)y-xy^(2)=0`
`implies(x-y)(x+y)+xy(x-y)=0`
`implies(x-y)(x+y+xy)=0`
`impliesx+y+xy=0`
`impliesy(1+x)= -x`
`impliesy=(-x)/(1+x)`
Differentiate both sides w.r.t. x
`(dy)/(dx)=((1+x)(d)/(dx)(-x)-(-x)(d)/(dx)(1+x))/((1+x)^(2))`
`=((1+x)(-1)+x(1))/((1+x)^(2))=(-1)/((1+x)^(2)) " " ` Hence proved.