Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 y^{2} \log{\left(x \right)} + 2 e^{x^{3}} \sin{\left(y^{3} \right)}=-9

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 6 x^{2} e^{x^{3}} \sin{\left(y^{3} \right)} + 6 y^{2} y' e^{x^{3}} \cos{\left(y^{3} \right)} + 12 y y' \log{\left(x \right)} + \frac{6 y^{2}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{x^{3} e^{x^{3}} \sin{\left(y^{3} \right)} + y^{2}}{x y \left(y e^{x^{3}} \cos{\left(y^{3} \right)} + 2 \log{\left(x \right)}\right)}