Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 4 e^{x^{3}} \log{\left(y \right)} - 4 \sin{\left(x^{3} \right)} \sin{\left(y^{3} \right)}=-44$

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