Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{2} \sqrt{y} \log{\left(x \right)} - 9 e^{x^{3}} e^{y^{3}}=33

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 27 x^{2} e^{x^{3}} e^{y^{3}} - 27 y^{2} y' e^{x^{3}} e^{y^{3}} + \frac{3 \sqrt{2} y' \log{\left(x \right)}}{\sqrt{y}} + \frac{6 \sqrt{2} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 9 x^{3} \sqrt{y} e^{x^{3} + y^{3}} + 2 \sqrt{2} y}{x \left(9 y^{\frac{5}{2}} e^{x^{3} + y^{3}} - \sqrt{2} \log{\left(x \right)}\right)}