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
$\frac{d y}{d x} =\frac{(x-1) \frac{d}{d x}(x+1)-(x+1) \frac{d}{d x}(x-1)}{(x-1)^{2}}$
$=\frac{(x-1)(1+0)-(x+1)(1-0)}{(x-1)^{2}}=\frac{x-1-x-1}{(x-1)^{2}}$
$\Rightarrow \frac{d y}{d x} =\frac{-2}{(x-1)^{2}}=\frac{-2}{(1-x)^{2}}$