Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 4 y^{3} \log{\left(x \right)} - 4 e^{x^{3}} \cos{\left(y^{3} \right)}=-20

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