Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 x^{2} \sin{\left(y \right)} + 9 y^{2} \log{\left(x \right)}=-38

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 x^{2} y' \cos{\left(y \right)} - 4 x \sin{\left(y \right)} + 18 y y' \log{\left(x \right)} + \frac{9 y^{2}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 4 x^{2} \sin{\left(y \right)} + 9 y^{2}}{2 x \left(x^{2} \cos{\left(y \right)} - 9 y \log{\left(x \right)}\right)}