Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 \sqrt{5} \sqrt{x} e^{y^{2}} + 9 y^{3} \sin{\left(x^{3} \right)}=3

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 16 \sqrt{5} \sqrt{x} y y' e^{y^{2}} + 27 x^{2} y^{3} \cos{\left(x^{3} \right)} + 27 y^{2} y' \sin{\left(x^{3} \right)} - \frac{4 \sqrt{5} e^{y^{2}}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 27 x^{\frac{5}{2}} y^{3} \cos{\left(x^{3} \right)} + 4 \sqrt{5} e^{y^{2}}}{y \left(27 \sqrt{x} y \sin{\left(x^{3} \right)} - 16 \sqrt{5} x e^{y^{2}}\right)}