Question:
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):
$\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$
Solution:
Let $f(x)=\frac{1+\frac{1}{x}}{1-\frac{1}{x}}=\frac{\frac{x+1}{x-1}}{\frac{x-1}{x}}=\frac{x+1}{x-1}$, where $x \neq 0$
By quotient rule,
$f^{\prime}(x)=\frac{(x-1) \frac{d}{d x}(x+1)-(x+1) \frac{d}{d x}(x-1)}{(x-1)^{2}}, x \neq 0,1$
$=\frac{(x-1)(1)-(x+1)(1)}{(x-1)^{2}}, x \neq 0,1$
$=\frac{x-1-x-1}{(x-1)^{2}}, x \neq 0,1$
$=\frac{-2}{(x-1)^{2}}, x \neq 0,1$
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