Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 18 \sqrt{2} \sqrt{x} y^{2} + 12 \sqrt{2} x^{2} \sqrt{y}=2

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 36 \sqrt{2} \sqrt{x} y y' + \frac{6 \sqrt{2} x^{2} y'}{\sqrt{y}} + 24 \sqrt{2} x \sqrt{y} - \frac{9 \sqrt{2} y^{2}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 8 x^{\frac{3}{2}} y + 3 y^{\frac{5}{2}}}{2 \left(x^{\frac{5}{2}} - 6 x y^{\frac{3}{2}}\right)}