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