Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 2 \sqrt{7} \sqrt{x} \log{\left(y \right)} + 8 \sqrt{7} \sqrt{y} \log{\left(x \right)}=27$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: \frac{2 \sqrt{7} \sqrt{x} y'}{y} + \frac{4 \sqrt{7} y' \log{\left(x \right)}}{\sqrt{y}} + \frac{8 \sqrt{7} \sqrt{y}}{x} + \frac{\sqrt{7} \log{\left(y \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{8 \sqrt{x} y^{2} + x y^{\frac{3}{2}} \log{\left(y \right)}}{4 x^{\frac{3}{2}} y \log{\left(x \right)} + 2 x^{2} \sqrt{y}}$