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

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