Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \sqrt{x} \log{\left(y \right)} - 5 x^{3} \sqrt{y}=-3$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{6 \sqrt{x} y'}{y} - \frac{5 x^{3} y'}{2 \sqrt{y}} - 15 x^{2} \sqrt{y} - \frac{3 \log{\left(y \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{30 x^{\frac{5}{2}} y^{2} + 6 y^{\frac{3}{2}} \log{\left(y \right)}}{5 x^{\frac{7}{2}} y + 12 x \sqrt{y}}$