Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 2 y \cos{\left(x^{3} \right)} + 5 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-31$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 6 x^{2} y \sin{\left(x^{3} \right)} + 15 x^{2} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + 15 y^{2} y' \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 2 y' \cos{\left(x^{3} \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{x^{2} \left(6 y \sin{\left(x^{3} \right)} - 15 \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)}\right)}{15 y^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 2 \cos{\left(x^{3} \right)}}$