Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 15 \sqrt{x} \sqrt{y} + 6 x^{2} \log{\left(y \right)}=-30

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{15 \sqrt{x} y'}{2 \sqrt{y}} + \frac{6 x^{2} y'}{y} + 12 x \log{\left(y \right)} + \frac{15 \sqrt{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{8 x^{\frac{3}{2}} y^{\frac{3}{2}} \log{\left(y \right)} + 5 y^{2}}{4 x^{\frac{5}{2}} \sqrt{y} + 5 x y}