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# Find the derivative of the following functions ( it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers ) $\large\frac{1+\Large\frac{1}{x}}{1-\Large\frac{1}{x}}$

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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$
answered Apr 12, 2014

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