Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 8 y^{3} \sin{\left(x^{2} \right)} - 9 \log{\left(x \right)} \cos{\left(y \right)}=20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 16 x y^{3} \cos{\left(x^{2} \right)} + 24 y^{2} y' \sin{\left(x^{2} \right)} + 9 y' \log{\left(x \right)} \sin{\left(y \right)} - \frac{9 \cos{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 16 x^{2} y^{3} \cos{\left(x^{2} \right)} + 9 \cos{\left(y \right)}}{3 x \left(8 y^{2} \sin{\left(x^{2} \right)} + 3 \log{\left(x \right)} \sin{\left(y \right)}\right)}