Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle y = \ln{\left(x y \right)}

Taking the derivative gives LaTeX: \displaystyle y' = \frac{1}{x y}\cdot (xy'+y) . Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \left\{\frac{y}{x \left(y - 1\right)}\right\}