Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 e^{x^{3}} e^{y^{3}} - 8 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=17

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 x^{2} e^{x^{3}} e^{y^{3}} - 24 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 18 y^{2} y' e^{x^{3}} e^{y^{3}} - 24 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{x^{2} \left(3 e^{x^{3} + y^{3}} + 4 \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{y^{2} \left(3 e^{x^{3} + y^{3}} + 4 \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)}\right)}