# How do you find the derivative of y= ln (x/(x-1))?

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{x \left(x - 1\right)}$
$y = \ln \left(\frac{x}{x - 1}\right)$
$\therefore y = \ln \left(x\right) - \ln \left(x - 1\right)$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x} - \frac{1}{x - 1}$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left(x - 1\right) - x}{x \left(x - 1\right)}$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{x \left(x - 1\right)}$