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 nonzero constants and m and n are integers): $\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$

Answer 1

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≠0
By the quotient rule,
f'(x)= (x-1) $\frac{d}{dx}$ (x+1) - (x+1)\frac{d}{dx}$$\frac{(x-1)}{(x-1)^2}. x≠0, 1
=$\frac{(x-1)(1)-(x+1)(1)}{(x-1)^2}$, x≠0,1
=$\frac{x-1-x-1}{(x-1)^2}$, x≠0,1
=$-\frac{2}{(x-1)^2}$, x≠0,1