Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 8 \sqrt{6} \sqrt{x} y^{3} + 6 \sqrt{6} \sqrt{y} e^{x^{2}}=-41$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 24 \sqrt{6} \sqrt{x} y^{2} y' + 12 \sqrt{6} x \sqrt{y} e^{x^{2}} + \frac{3 \sqrt{6} y' e^{x^{2}}}{\sqrt{y}} + \frac{4 \sqrt{6} y^{3}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{12 x^{\frac{3}{2}} y e^{x^{2}} + 4 y^{\frac{7}{2}}}{3 \sqrt{x} e^{x^{2}} + 24 x y^{\frac{5}{2}}}$