Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 x^{3} \cos{\left(y^{3} \right)} + 8 e^{y} \log{\left(x \right)}=-44

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