Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 e^{x^{3}} \sin{\left(y^{2} \right)} + 2 \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)}=-31

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 9 x^{2} e^{x^{3}} \sin{\left(y^{2} \right)} + 4 x \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 6 y y' e^{x^{3}} \cos{\left(y^{2} \right)} - 4 y y' \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x \left(- 9 x e^{x^{3}} \sin{\left(y^{2} \right)} + 4 \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)}\right)}{2 y \left(3 e^{x^{3}} \cos{\left(y^{2} \right)} + 2 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}\right)}