Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 x^{3} y^{3} + 5 \log{\left(x \right)} \sin{\left(y \right)}=31

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