Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 36 \sqrt{x} \sqrt{y} + 2 \log{\left(y \right)} \sin{\left(x \right)}=3

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{18 \sqrt{x} y'}{\sqrt{y}} + 2 \log{\left(y \right)} \cos{\left(x \right)} + \frac{2 y' \sin{\left(x \right)}}{y} - \frac{18 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- \sqrt{x} y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x \right)} + 9 y^{2}}{\sqrt{x} \sqrt{y} \sin{\left(x \right)} - 9 x y}