Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 4 x^{3} e^{y^{3}} + \sqrt{7} \sqrt{y} e^{x^{3}}=-50$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 12 x^{3} y^{2} y' e^{y^{3}} + 3 \sqrt{7} x^{2} \sqrt{y} e^{x^{3}} - 12 x^{2} e^{y^{3}} + \frac{\sqrt{7} y' e^{x^{3}}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{6 x^{2} \left(- 4 \sqrt{y} e^{y^{3}} + \sqrt{7} y e^{x^{3}}\right)}{24 x^{3} y^{\frac{5}{2}} e^{y^{3}} - \sqrt{7} e^{x^{3}}}$