Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle y = e^{x y}

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