Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 x^{2} y^{2} - 9 \log{\left(x \right)} \sin{\left(y^{2} \right)}=31

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