Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 e^{x^{3}} e^{y^{3}} - 4 \sin{\left(x^{2} \right)} \cos{\left(y^{2} \right)}=-37

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 27 x^{2} e^{x^{3}} e^{y^{3}} - 8 x \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} + 27 y^{2} y' e^{x^{3}} e^{y^{3}} + 8 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(- 27 x e^{x^{3} + y^{3}} + 8 \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)}\right)}{y \left(27 y e^{x^{3} + y^{3}} + 8 \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)}\right)}