#### Answer

$$\frac{dy}{dx}=-\frac{y}{x+2y}$$
$$\frac{d^2y}{dx^2}=\frac{2xy+2y^2}{(x+2y)^3}$$

#### Work Step by Step

$$xy+y^2=1$$
1) Find $dy/dx$:
Differentiate both sides of the equation with respect to $x$: $$\frac{d}{dx}(xy)+\frac{d}{dx}(y^2)=\frac{d}{dx}(1)$$ $$y+x\frac{dy}{dx}+2y\frac{dy}{dx}=0$$
Isolate the terms with $dy/dx$ into one side and solve for $dy/dx$: $$x\frac{dy}{dx}+2y\frac{dy}{dx}=-y$$ $$(x+2y)\frac{dy}{dx}=-y$$ $$\frac{dy}{dx}=-\frac{y}{x+2y}$$
2) Find $d^2y/dx^2$:
Differentiate the derivative $dy/dx$ as normal, using Quotient Rule: $$\frac{d^2y}{dx^2}=\frac{d}{dx}\Big(-\frac{y}{x+2y}\Big)=-\frac{(x+2y)\frac{dy}{dx}-y\frac{d}{dx}(x+2y)}{(x+2y)^2}$$ $$\frac{d^2y}{dx^2}=-\frac{(x+2y)\frac{dy}{dx}-y(1+2\frac{dy}{dx})}{(x+2y)^2}$$
Now we can use the result $dy/dx=-y/(x+2y)$ from part 1) to do substitution here:
$$\frac{d^2y}{dx^2}=-\frac{(x+2y)\Big(-\frac{y}{x+2y}\Big)-y\Big(1-\frac{2y}{x+2y}\Big)}{(x+2y)^2}$$ $$\frac{d^2y}{dx^2}=-\Bigg[\frac{-y-y+\frac{2y^2}{x+2y}}{(x+2y)^2}\Bigg]=\frac{2y-\frac{2y^2}{x+2y}}{(x+2y)^2}=\frac{\frac{2y(x+2y)-2y^2}{x+2y}}{(x+2y)^2}$$ $$\frac{d^2y}{dx^2}=\frac{2xy+2y^2}{(x+2y)^3}$$