Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 y^{2} \cos{\left(x \right)} - 6 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-18

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 18 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 9 y^{2} \sin{\left(x \right)} - 18 y y' \cos{\left(x \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + \frac{y^{2} \sin{\left(x \right)}}{2}}{y \left(y \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + \cos{\left(x \right)}\right)}