Question

Find the derivative of $\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$.

Answer 1

Let $f\left(x\right)=\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$
$\Rightarrow f\left(x\right)=\frac{\frac{x+1}{x}}{\frac{x-1}{x}}$
$\Rightarrow f\left(x\right)=\frac{x+1}{x-1}$

Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{\left[\right(x-1\left)\frac{d}{dx}\right(x+1)-(x+1\left)\frac{d}{dx}\right(x-1\left)\right]}{(x-1{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1(x-1)-1(x+1)}{(x-1{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{x-1-x-1}{(x-1{)}^2}$
$\therefore f^{\prime }\left(x\right)=\frac{-2}{(x-1{)}^2}$