Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle y = \tan^{-1}{\left(x y \right)}

Taking the derivative gives LaTeX: \displaystyle y' = \frac{1}{x^{2} y^{2} + 1}\cdot (xy'+y) . Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \left\{\frac{y}{x^{2} y^{2} - x + 1}\right\}