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)} - 2 e^{y^{3}} \sin{\left(x^{3} \right)}=7

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