Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 7 x^{2} y^{2} - 4 \sqrt{2} \sqrt{y} \sin{\left(x \right)}=8$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 14 x^{2} y y' - 14 x y^{2} - 4 \sqrt{2} \sqrt{y} \cos{\left(x \right)} - \frac{2 \sqrt{2} y' \sin{\left(x \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{7 x y^{\frac{5}{2}} + 2 \sqrt{2} y \cos{\left(x \right)}}{7 x^{2} y^{\frac{3}{2}} + \sqrt{2} \sin{\left(x \right)}}$