Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 12 \sqrt{y} \log{\left(x \right)} - 3 \log{\left(y \right)} \cos{\left(x \right)}=9

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