Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - x^{2} e^{y^{3}} + 2 \log{\left(x \right)} \log{\left(y \right)}=-33

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