Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 y^{2} \log{\left(x \right)} + 2 \cos{\left(x^{3} \right)} \cos{\left(y^{3} \right)}=-40

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