Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 2 y e^{x^{3}} - 7 e^{y^{3}} \sin{\left(x \right)}=6$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 6 x^{2} y e^{x^{3}} - 21 y^{2} y' e^{y^{3}} \sin{\left(x \right)} - 2 y' e^{x^{3}} - 7 e^{y^{3}} \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{6 x^{2} y e^{x^{3}} + 7 e^{y^{3}} \cos{\left(x \right)}}{21 y^{2} e^{y^{3}} \sin{\left(x \right)} + 2 e^{x^{3}}}$