Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 18 \sqrt{2} \sqrt{x} e^{y} - 10 \sqrt{2} \sqrt{y} e^{x}=-25$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 18 \sqrt{2} \sqrt{x} y' e^{y} - 10 \sqrt{2} \sqrt{y} e^{x} - \frac{5 \sqrt{2} y' e^{x}}{\sqrt{y}} + \frac{9 \sqrt{2} e^{y}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 10 \sqrt{x} y e^{x} + 9 \sqrt{y} e^{y}}{5 \sqrt{x} e^{x} - 18 x \sqrt{y} e^{y}}$