Question

If $xy=x+y,$ then $\left(\frac{dy}{dx}\right)$ is equal to:

• A. $\frac{xy}{(1-x)}$
• B. $\frac{(1+y)}{(1-x)}$
• C. $\frac{y}{(1-xy)}$
• D. $\frac{-1}{{(x-1)}^2}$
• E. $\frac{1}{(x^2-1)}$

Answer 1

Answer: (D)

$xy=x+y\Rightarrow xy-y=x$ $\Rightarrow$ $y(x-1)=x$ $\Rightarrow$ $yx=x+y$ On differentiating w.r.t. $x,$ we get $x\frac{dy}{dx}+y=1+\frac{dy}{dx}$ $\Rightarrow$ $\frac{dy}{dx}(x-1)=1-y$ $\Rightarrow$ $\frac{dy}{dx}=\frac{1-y}{x-1}=\frac{1-\frac{x}{x-1}}{x-1}$ $=\frac{x-1-x}{{(x-1)}^2}=\frac{-1}{{(x-1)}^2}$