Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 e^{y^{2}} \sin{\left(x^{3} \right)} - 4 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=9

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 12 x^{2} e^{y^{2}} \cos{\left(x^{3} \right)} + 8 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} - 12 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 8 y y' e^{y^{2}} \sin{\left(x^{3} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x \left(- 3 x e^{y^{2}} \cos{\left(x^{3} \right)} + 2 \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)}\right)}{y \left(3 y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \sin{\left(x^{3} \right)}\right)}