Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 x^{3} \sin{\left(y^{3} \right)} + 6 \sqrt{5} \sqrt{y} \cos{\left(x^{2} \right)}=47$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 24 x^{3} y^{2} y' \cos{\left(y^{3} \right)} - 24 x^{2} \sin{\left(y^{3} \right)} - 12 \sqrt{5} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{3 \sqrt{5} y' \cos{\left(x^{2} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{4 x \left(2 x \sqrt{y} \sin{\left(y^{3} \right)} + \sqrt{5} y \sin{\left(x^{2} \right)}\right)}{8 x^{3} y^{\frac{5}{2}} \cos{\left(y^{3} \right)} - \sqrt{5} \cos{\left(x^{2} \right)}}$