I am trying to solve the problem: ${x}^{2}+xy+{y}^{3}=0$ using implicit differentiation.

My workings:

$(1)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}[{x}^{2}]\phantom{\rule{thinmathspace}{0ex}}+\frac{d}{dx}[xy]\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}[{y}^{3}]=\frac{d}{dx}[0]$

$(2)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{d{y}^{3}}{dy}\frac{dy}{dx}=0$

$(3)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}(\frac{dy}{dx})=0$

$(4)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=\overline{){\displaystyle -\frac{2x+y}{3{y}^{2}}}}$

But the answer says it should be:

$(3)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dxy}{dy}\frac{dy}{dx}+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}(\frac{dy}{dx})=0$

$(4)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dy}{dx}(x+3{y}^{2})=0$

$(5)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=-\frac{2x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}y}{x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}3{y}^{2}}$

Why?

My workings:

$(1)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}[{x}^{2}]\phantom{\rule{thinmathspace}{0ex}}+\frac{d}{dx}[xy]\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\frac{d}{dx}[{y}^{3}]=\frac{d}{dx}[0]$

$(2)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{d{y}^{3}}{dy}\frac{dy}{dx}=0$

$(3)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}(\frac{dy}{dx})=0$

$(4)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=\overline{){\displaystyle -\frac{2x+y}{3{y}^{2}}}}$

But the answer says it should be:

$(3)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dxy}{dy}\frac{dy}{dx}+3{y}^{2}\phantom{\rule{thinmathspace}{0ex}}(\frac{dy}{dx})=0$

$(4)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2x+y+\frac{dy}{dx}(x+3{y}^{2})=0$

$(5)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{dy}{dx}=-\frac{2x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}y}{x\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}3{y}^{2}}$

Why?