Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 \sqrt{3} \sqrt{x} \log{\left(y \right)} - 5 e^{x} e^{y}=31

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{9 \sqrt{3} \sqrt{x} y'}{y} - 5 y' e^{x} e^{y} - 5 e^{x} e^{y} + \frac{9 \sqrt{3} \log{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- 10 \sqrt{x} e^{x + y} + 9 \sqrt{3} \log{\left(y \right)}\right)}{2 \left(5 \sqrt{x} y e^{x + y} - 9 \sqrt{3} x\right)}