Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 \sqrt{5} \sqrt{x} \log{\left(y \right)} + 9 x^{3} y^{3}=-2

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{3 \sqrt{5} \sqrt{x} y'}{y} + 27 x^{3} y^{2} y' + 27 x^{2} y^{3} + \frac{3 \sqrt{5} \log{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{y \left(18 x^{\frac{5}{2}} y^{3} + \sqrt{5} \log{\left(y \right)}\right)}{18 x^{\frac{7}{2}} y^{3} + 2 \sqrt{5} x}