Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 6 x^{2} \sin{\left(y^{2} \right)} + \log{\left(x \right)} \cos{\left(y^{3} \right)}=-44

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 12 x^{2} y y' \cos{\left(y^{2} \right)} - 12 x \sin{\left(y^{2} \right)} - 3 y^{2} y' \log{\left(x \right)} \sin{\left(y^{3} \right)} + \frac{\cos{\left(y^{3} \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 12 x^{2} \sin{\left(y^{2} \right)} + \cos{\left(y^{3} \right)}}{3 x y \left(4 x^{2} \cos{\left(y^{2} \right)} + y \log{\left(x \right)} \sin{\left(y^{3} \right)}\right)}