Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 e^{x^{2}} e^{y^{2}} - 6 \cos{\left(x^{3} \right)} \cos{\left(y^{3} \right)}=-25

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 x^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 10 x e^{x^{2}} e^{y^{2}} + 18 y^{2} y' \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 10 y y' e^{x^{2}} e^{y^{2}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x \left(- 9 x \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 5 e^{x^{2} + y^{2}}\right)}{y \left(9 y \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 5 e^{x^{2} + y^{2}}\right)}