Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \sqrt{5} \sqrt{x} e^{y^{3}} - 8 \sqrt{5} \sqrt{y} \log{\left(x \right)}=4$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 18 \sqrt{5} \sqrt{x} y^{2} y' e^{y^{3}} - \frac{4 \sqrt{5} y' \log{\left(x \right)}}{\sqrt{y}} - \frac{8 \sqrt{5} \sqrt{y}}{x} - \frac{3 \sqrt{5} e^{y^{3}}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{8 \sqrt{x} y + 3 x \sqrt{y} e^{y^{3}}}{4 x^{\frac{3}{2}} \log{\left(x \right)} + 18 x^{2} y^{\frac{5}{2}} e^{y^{3}}}$