Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 e^{x} \cos{\left(y^{3} \right)} - \log{\left(x \right)} \log{\left(y \right)}=-30

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 27 y^{2} y' e^{x} \sin{\left(y^{3} \right)} - 9 e^{x} \cos{\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(9 x e^{x} \cos{\left(y^{3} \right)} + \log{\left(y \right)}\right)}{x \left(27 y^{3} e^{x} \sin{\left(y^{3} \right)} - \log{\left(x \right)}\right)}