Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 \sqrt{3} \sqrt{y} \log{\left(x \right)} - 3 e^{x} e^{y}=24

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