Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 32 \sqrt{x} \sqrt{y} + 6 x y=-3

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{16 \sqrt{x} y'}{\sqrt{y}} + 6 x y' + 6 y + \frac{16 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{3 \sqrt{x} y^{\frac{3}{2}} + 8 y}{3 x^{\frac{3}{2}} \sqrt{y} + 8 x}