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}} - y^{2} \log{\left(x \right)}=-11

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}} - 2 y y' \log{\left(x \right)} - \frac{y^{2}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 2 x^{2} e^{y^{3}} + y^{2}}{x y \left(3 x^{2} y e^{y^{3}} - 2 \log{\left(x \right)}\right)}