Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 6 \sqrt{2} \sqrt{y} \sin{\left(x^{3} \right)} + 5 y^{3} e^{x^{3}}=12$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 18 \sqrt{2} x^{2} \sqrt{y} \cos{\left(x^{3} \right)} + 15 x^{2} y^{3} e^{x^{3}} + 15 y^{2} y' e^{x^{3}} + \frac{3 \sqrt{2} y' \sin{\left(x^{3} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{x^{2} \left(5 y^{\frac{7}{2}} e^{x^{3}} + 6 \sqrt{2} y \cos{\left(x^{3} \right)}\right)}{5 y^{\frac{5}{2}} e^{x^{3}} + \sqrt{2} \sin{\left(x^{3} \right)}}$