Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 x \cos{\left(y^{3} \right)} - 4 e^{y^{2}} \cos{\left(x^{3} \right)}=-41

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + 9 x y^{2} y' \sin{\left(y^{3} \right)} - 8 y y' e^{y^{2}} \cos{\left(x^{3} \right)} - 3 \cos{\left(y^{3} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 \left(- 4 x^{2} e^{y^{2}} \sin{\left(x^{3} \right)} + \cos{\left(y^{3} \right)}\right)}{y \left(9 x y \sin{\left(y^{3} \right)} - 8 e^{y^{2}} \cos{\left(x^{3} \right)}\right)}