Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - e^{x^{2}} \sin{\left(y^{3} \right)} - 8 \log{\left(y \right)} \sin{\left(x^{3} \right)}=45$

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