Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 10 \sqrt{2} \sqrt{x} \cos{\left(y^{3} \right)} - 6 y e^{x}=22

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 30 \sqrt{2} \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} - 6 y e^{x} - 6 y' e^{x} + \frac{5 \sqrt{2} \cos{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- \sqrt{x} y e^{x} + \frac{5 \sqrt{2} \cos{\left(y^{3} \right)}}{6}}{\sqrt{x} e^{x} + 5 \sqrt{2} x y^{2} \sin{\left(y^{3} \right)}}