Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 16 \sqrt{y} \log{\left(x \right)} + 8 e^{x^{3}} \sin{\left(y^{3} \right)}=-21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 24 x^{2} e^{x^{3}} \sin{\left(y^{3} \right)} + 24 y^{2} y' e^{x^{3}} \cos{\left(y^{3} \right)} - \frac{8 y' \log{\left(x \right)}}{\sqrt{y}} - \frac{16 \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 3 x^{3} \sqrt{y} e^{x^{3}} \sin{\left(y^{3} \right)} + 2 y}{x \left(3 y^{\frac{5}{2}} e^{x^{3}} \cos{\left(y^{3} \right)} - \log{\left(x \right)}\right)}