Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& \text{ Given, }x\sqrt{1+y}+y\sqrt{1+x}=0\\ & \Rightarrow x\sqrt{1+y}=-y\sqrt{1+x}\\ & \Rightarrow (x\sqrt{1+y}{)}^2=(-y\sqrt{1+x}{)}^2\end{array}$ $\begin{array}{lr}\Rightarrow & x^2(1+y)=y^2(1+x)\\ \Rightarrow & x^2-y^2=y^{2}x-x^{2}y\\ \Rightarrow & (x-y)(x+y)=xy(y-x)\\ \Rightarrow & x+y=-xy\\ \Rightarrow & x+y+xy=0\\ \Rightarrow & y=\frac{-x}{1+x}=-1+\frac{1}{1+x}\end{array}$ Differentiating w.r.t, $x$ $\frac{dy}{dx}=-\frac{1}{(1+x{)}^2}$