Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 2 x e^{y^{3}} - 14 \sqrt{y} e^{x^{3}}=-19$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 42 x^{2} \sqrt{y} e^{x^{3}} - 6 x y^{2} y' e^{y^{3}} - 2 e^{y^{3}} - \frac{7 y' e^{x^{3}}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{42 x^{2} y e^{x^{3}} + 2 \sqrt{y} e^{y^{3}}}{6 x y^{\frac{5}{2}} e^{y^{3}} + 7 e^{x^{3}}}$