Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 x^{3} y^{3} + \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)}=-10

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