Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 \sqrt{7} \sqrt{y} \log{\left(x \right)} - 5 e^{x} e^{y}=36

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 5 y' e^{x} e^{y} - 5 e^{x} e^{y} - \frac{3 \sqrt{7} y' \log{\left(x \right)}}{2 \sqrt{y}} - \frac{3 \sqrt{7} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{10 x \sqrt{y} e^{x + y} + 6 \sqrt{7} y}{x \left(10 \sqrt{y} e^{x + y} + 3 \sqrt{7} \log{\left(x \right)}\right)}