Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 y \sin{\left(x^{3} \right)} - 6 e^{x} \cos{\left(y^{3} \right)}=-23

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 27 x^{2} y \cos{\left(x^{3} \right)} + 18 y^{2} y' e^{x} \sin{\left(y^{3} \right)} + 9 y' \sin{\left(x^{3} \right)} - 6 e^{x} \cos{\left(y^{3} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 9 x^{2} y \cos{\left(x^{3} \right)} + 2 e^{x} \cos{\left(y^{3} \right)}}{3 \left(2 y^{2} e^{x} \sin{\left(y^{3} \right)} + \sin{\left(x^{3} \right)}\right)}