Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 7 \sqrt{3} \sqrt{y} e^{x^{3}} - 2 e^{y^{3}} \cos{\left(x \right)}=37$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 21 \sqrt{3} x^{2} \sqrt{y} e^{x^{3}} - 6 y^{2} y' e^{y^{3}} \cos{\left(x \right)} + 2 e^{y^{3}} \sin{\left(x \right)} - \frac{7 \sqrt{3} y' e^{x^{3}}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 \left(- 21 \sqrt{3} x^{2} y e^{x^{3}} + 2 \sqrt{y} e^{y^{3}} \sin{\left(x \right)}\right)}{12 y^{\frac{5}{2}} e^{y^{3}} \cos{\left(x \right)} + 7 \sqrt{3} e^{x^{3}}}$