Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 10 \sqrt{2} \sqrt{y} \sin{\left(x^{3} \right)} + 4 e^{y} \cos{\left(x \right)}=-2$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 30 \sqrt{2} x^{2} \sqrt{y} \cos{\left(x^{3} \right)} + 4 y' e^{y} \cos{\left(x \right)} - 4 e^{y} \sin{\left(x \right)} - \frac{5 \sqrt{2} y' \sin{\left(x^{3} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 \left(15 \sqrt{2} x^{2} y \cos{\left(x^{3} \right)} + 2 \sqrt{y} e^{y} \sin{\left(x \right)}\right)}{4 \sqrt{y} e^{y} \cos{\left(x \right)} - 5 \sqrt{2} \sin{\left(x^{3} \right)}}$