Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 5 \sqrt{6} \sqrt{x} \log{\left(y \right)} - 9 \sqrt{6} x^{2} \sqrt{y}=-41$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{5 \sqrt{6} \sqrt{x} y'}{y} - \frac{9 \sqrt{6} x^{2} y'}{2 \sqrt{y}} - 18 \sqrt{6} x \sqrt{y} - \frac{5 \sqrt{6} \log{\left(y \right)}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{36 x^{\frac{3}{2}} y^{2} + 5 y^{\frac{3}{2}} \log{\left(y \right)}}{9 x^{\frac{5}{2}} y + 10 x \sqrt{y}}$