Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 4 \sqrt{5} \sqrt{x} y^{3} + 4 e^{x^{3}} \sin{\left(y \right)}=-41$

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