**Toolbox:**

- Shift one term to R.H.S square and then differentiate

Step 1

$ x \sqrt {1+y}=-y\sqrt{1+x}$ squaring both sides

Step 2

$ \Rightarrow x^2(1+y)=y^2(1+x)$

$ \Rightarrow x^2 + x^2y-y^2-yx^2=0$

Step 3

$ (x-y)(x+y+xy)=0$

$ \Rightarrow x+y+xy = 0 $

$ y = \frac{-x}{1+x}$ differentiate both sides to get the answer.