Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 e^{x} \log{\left(y \right)} - 3 e^{y} \log{\left(x \right)}=-50

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 3 y' e^{y} \log{\left(x \right)} + 4 e^{x} \log{\left(y \right)} + \frac{4 y' e^{x}}{y} - \frac{3 e^{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(4 x e^{x} \log{\left(y \right)} - 3 e^{y}\right)}{x \left(3 y e^{y} \log{\left(x \right)} - 4 e^{x}\right)}