Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 y^{2} \cos{\left(x \right)} + 3 \log{\left(x \right)} \sin{\left(y^{3} \right)}=-12

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