Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - y^{2} e^{x} + 4 \log{\left(x \right)} \log{\left(y \right)}=-16

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - y^{2} e^{x} - 2 y y' e^{x} + \frac{4 y' \log{\left(x \right)}}{y} + \frac{4 \log{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- x y^{2} e^{x} + 4 \log{\left(y \right)}\right)}{2 x \left(y^{2} e^{x} - 2 \log{\left(x \right)}\right)}