Find the derivative of the following fucctions (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) :
$\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$

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Solution

Let $f (x) = \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}} = \frac{\frac{x + 1}{x}}{\frac{x - 1}{x}} = \frac{x + 1}{x - 1}$
where $x \neq 0$
Thus using quotient rule
$f^{'} (x) = \frac{(x - 1) \frac{d}{d x} (x + 1) - (x + 1) \frac{d}{d x} (x - 1)}{{(x - 1)}^{2}}$
$= \frac{x - 1 - (x + 1)}{(x - 1)^{2}} = \frac{- 2}{(x - 1)^{2}}$

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