Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 y^{3} \cos{\left(x^{2} \right)} - 2 \log{\left(x \right)} \sin{\left(y^{3} \right)}=32

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 4 x y^{3} \sin{\left(x^{2} \right)} - 6 y^{2} y' \log{\left(x \right)} \cos{\left(y^{3} \right)} - 6 y^{2} y' \cos{\left(x^{2} \right)} - \frac{2 \sin{\left(y^{3} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 x^{2} y^{3} \sin{\left(x^{2} \right)} - \sin{\left(y^{3} \right)}}{3 x y^{2} \left(\log{\left(x \right)} \cos{\left(y^{3} \right)} + \cos{\left(x^{2} \right)}\right)}