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 \(y' = \left(\log _{2}\left(\frac{x+7}{x-7}\right)\right)^{\ln 2}\left(\frac{x-7}{x-7}-\frac{x+7}{(x-7)^2}\right)\ln 2\left(\frac{1}{x+7}\log _{2}\left(\frac{x+7}{x-7}\right)\right)\)

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 \(\boxed{\left(\log _{2}\left(\frac{x+7}{x-7}\right)\right)^{\ln 2}\left(\frac{x-7}{x-7}-\frac{x+7}{(x-7)^2}\right)\ln 2\left(\frac{1}{x+7}\log _{2}\left(\frac{x+7}{x-7}\right)\right)}\)