Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 \sqrt{6} \sqrt{x} \log{\left(y \right)} - 9 \sqrt{6} \sqrt{y} \sin{\left(x^{3} \right)}=-8

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{4 \sqrt{6} \sqrt{x} y'}{y} - 27 \sqrt{6} x^{2} \sqrt{y} \cos{\left(x^{3} \right)} - \frac{9 \sqrt{6} y' \sin{\left(x^{3} \right)}}{2 \sqrt{y}} - \frac{2 \sqrt{6} \log{\left(y \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{54 x^{\frac{5}{2}} y^{2} \cos{\left(x^{3} \right)} + 4 y^{\frac{3}{2}} \log{\left(y \right)}}{9 \sqrt{x} y \sin{\left(x^{3} \right)} + 8 x \sqrt{y}}