Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 y e^{x} + \log{\left(x \right)} \log{\left(y \right)}=3

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 9 y e^{x} - 9 y' e^{x} + \frac{y' \log{\left(x \right)}}{y} + \frac{\log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 9 x y e^{x} + \log{\left(y \right)}\right)}{x \left(9 y e^{x} - \log{\left(x \right)}\right)}