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

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 8 \sqrt{2} \sqrt{x} y' \cos{\left(y \right)} - 9 x^{2} y^{3} \sin{\left(x^{3} \right)} + 9 y^{2} y' \cos{\left(x^{3} \right)} - \frac{4 \sqrt{2} \sin{\left(y \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{9 x^{\frac{5}{2}} y^{3} \sin{\left(x^{3} \right)} + 4 \sqrt{2} \sin{\left(y \right)}}{9 \sqrt{x} y^{2} \cos{\left(x^{3} \right)} - 8 \sqrt{2} x \cos{\left(y \right)}}$