Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 2 \sqrt{3} \sqrt{y} \cos{\left(x \right)} - 8 y^{3} e^{x^{3}}=41$

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