Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 2 x^{3} \sin{\left(y^{3} \right)} + \log{\left(x \right)} \log{\left(y \right)}=-41

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