Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle \sqrt{3} \sqrt{y} \log{\left(x \right)} + 5 e^{x} e^{y}=-39

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