Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{2} \sqrt{y} \cos{\left(x^{2} \right)} - 4 y^{3} e^{x}=-6

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 24 \sqrt{2} x \sqrt{y} \sin{\left(x^{2} \right)} - 4 y^{3} e^{x} - 12 y^{2} y' e^{x} + \frac{6 \sqrt{2} y' \cos{\left(x^{2} \right)}}{\sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(6 \sqrt{2} x y \sin{\left(x^{2} \right)} + y^{\frac{7}{2}} e^{x}\right)}{3 \left(- 2 y^{\frac{5}{2}} e^{x} + \sqrt{2} \cos{\left(x^{2} \right)}\right)}