Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 \sqrt{x} y - 4 e^{x} \sin{\left(y^{3} \right)}=-18

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