Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 8 \sqrt{5} x \sqrt{y} + 9 e^{x^{3}} e^{y^{3}}=-50$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 27 x^{2} e^{x^{3}} e^{y^{3}} + \frac{4 \sqrt{5} x y'}{\sqrt{y}} + 8 \sqrt{5} \sqrt{y} + 27 y^{2} y' e^{x^{3}} e^{y^{3}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{27 x^{2} \sqrt{y} e^{x^{3} + y^{3}} + 8 \sqrt{5} y}{4 \sqrt{5} x + 27 y^{\frac{5}{2}} e^{x^{3} + y^{3}}}$