Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 6 \sqrt{7} \sqrt{x} \sin{\left(y^{3} \right)} + 3 \log{\left(x \right)} \log{\left(y \right)}=-31$

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