If `y=x/(x+2)` , prove that `x(dy)/(dx)=(1-y)y`

Explanation

We have, `{y}=frac{x}{x+2}`

`frac{d y}{d x}=frac{d}{d x}(frac{x}{x+2}) `

`Rightarrow frac{d y}{d x}=frac{(x+2) frac{d}{d x}(x)-x frac{d}{d x}(x+2)}{(x+2)^{2}} `

`Rightarrow frac{d y}{d x}=frac{x+2-x}{(x+2)^{2}} `

`Rightarrow frac{d y}{d x}=frac{x+2}{(x+2)^{2}}-frac{x}{(x+2)^{2}} `

`Rightarrow frac{d y}{d x}=frac{1}{x+2}-frac{x y^{2}}{x^{2}}[because x+2=frac{x}{y}] `

`Rightarrow frac{d y}{d x}=frac{y}{x}-frac{y^{2}}{x} `

`Rightarrow frac{d y}{d x}=frac{1}{x} y(1-y) `

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