Question

The derivative of $\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$ is

• A. $\frac{2}{(1+x{)}^2}$
• B. $\frac{-2}{(1-x{)}^2}$
• C. $\frac{-1}{(1-x{)}^2}$
• D. $\frac{3}{(1-x{)}^2}$

Answer 1

Answer: (B)

Let $y=\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$
$\Rightarrow y=\frac{x+1}{x-1}$
Differentiating y w.r.t. x, we get
$\frac{dy}{dx}=\frac{(x-1)\frac{d}{dx}(x+1)-(x+1)\frac{d}{dx}(x-1)}{(x-1{)}^2}$
$=\frac{(x-1)(1+0)-(x+1)(1-0)}{(x-1{)}^2}=\frac{x-1-x-1}{(x-1{)}^2}$
$\Rightarrow \frac{dy}{dx}=\frac{-2}{(x-1{)}^2}=\frac{-2}{(1-x{)}^2}$