Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 \sqrt{2} x^{2} \sqrt{y} - 6 \sin{\left(x \right)} \sin{\left(y \right)}=-31$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - \frac{4 \sqrt{2} x^{2} y'}{\sqrt{y}} - 16 \sqrt{2} x \sqrt{y} - 6 y' \sin{\left(x \right)} \cos{\left(y \right)} - 6 \sin{\left(y \right)} \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{8 \sqrt{2} x y + 3 \sqrt{y} \sin{\left(y \right)} \cos{\left(x \right)}}{2 \sqrt{2} x^{2} + 3 \sqrt{y} \sin{\left(x \right)} \cos{\left(y \right)}}$