Find the derivative of $y$ with respect to $x$.
$$y={\log }_2\left({\left(\frac{x+7}{x-7}\right)}^{\ln 2}\right)$$

Step 1 ：Given the function $y={\log }_2\left({\left(\frac{x+7}{x-7}\right)}^{\ln 2}\right)$

Step 2 ：First, we simplify the equation using logarithmic properties. We use the property of logarithms that says ${\log }_b(a^c)=c{\log }_b(a)$ to simplify the equation.

Step 3 ：Next, we use the chain rule to find the derivative. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 4 ：Applying the chain rule, we get $y^{\prime }={({\log }_2\left(\frac{x+7}{x-7}\right))}^{\ln 2}(\frac{x-7}{x-7}-\frac{x+7}{(x-7{)}^2})\ln 2(\frac{1}{x+7}{\log }_2\left(\frac{x+7}{x-7}\right))$

Step 5 ：Final Answer: The derivative of the function $y={\log }_2\left({\left(\frac{x+7}{x-7}\right)}^{\ln 2}\right)$ with respect to $x$ is $\boxed{{\displaystyle {({\log }_2\left(\frac{x+7}{x-7}\right))}^{\ln 2}(\frac{x-7}{x-7}-\frac{x+7}{(x-7{)}^2})\ln 2(\frac{1}{x+7}{\log }_2\left(\frac{x+7}{x-7}\right))}}$