Single Variable Calculus, Chapter 7, 7.2-2, Section 7.2-2, Problem 24

Differentiate $\displaystyle f(x) = \frac{1 + \ln x}{1 - \ln x}$


$\begin{equation} \begin{aligned} \text{if } f(x) =& \frac{1 + \ln x}{1 - \ln x}, \text{ then by using Quotient Rule} \\ \\ f'(x) =& \frac{\displaystyle (1 - \ln x) \cdot \frac{d}{dx} (1 + \ln x) - (1 + \ln x) \cdot \frac{d}{dx} (1 - \ln x) }{(1 - \ln x)^2} \\ \\ f'(x) =& \frac{\displaystyle (1 - \ln x) \left( \frac{1}{x} \right) - (1 + \ln x) \left( \frac{-1}{x} \right)}{(1 - \ln x)^2} \\ \\ f'(x) =& \frac{\displaystyle \frac{1}{x} (1 - \ln x + 1 + \ln x) }{(1 - \ln x)^2} \\ \\ f'(x) =& \frac{2}{x(1 - \ln x)^2} \end{aligned} \end{equation}$