Let $f(x) = \large\frac{1+\Large\frac{1}{x}}{1-\Large\frac{1}{x}}$$ =\large\frac{\Large\frac{x+1}{x}}{\Large\frac{x-1}{x}}$$ = \large\frac{x+1}{x-1}$, where $ x \neq 0$

By quotient rule,

$ f'(x) = \large\frac{(x-1)\large\frac{d}{dx}(x+1)-(x+1) \large\frac{d}{dx}(x-1)}{(x-1)^2}$$, x \neq 0,1$

$ = \large\frac{(x-1)(1)-(x+1)(1)}{(x-1)^2}$$, x \neq 0,1$

$= \large\frac{x-1-x-1}{(x-1)^2}$$, x \neq 0, 1$

$= \large\frac{-2}{(x-1)^2}$$, x \neq 0, 1$