Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 \sqrt{3} x \sqrt{y} + 9 y \log{\left(x \right)}=-16

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{2 \sqrt{3} x y'}{\sqrt{y}} + 4 \sqrt{3} \sqrt{y} + 9 y' \log{\left(x \right)} + \frac{9 y}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{4 \sqrt{3} x y + 9 y^{\frac{3}{2}}}{x \left(2 \sqrt{3} x + 9 \sqrt{y} \log{\left(x \right)}\right)}