Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 e^{x^{2}} \cos{\left(y^{3} \right)} + 2 e^{y^{2}} \cos{\left(x^{3} \right)}=11

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