Find the derivative of the function. \[ y=\log _{7}\left(\sqrt{x^{2}-9}\right) \] \( y=\square \)

To find the derivative of the function \( y=\log _{7}\left(\sqrt{x^{2}-9}\right) \), we first rewrite the logarithmic function using the change of base formula: \[ y = \frac{\log\left(\sqrt{x^{2}-9}\right)}{\log(7)}. \] Now, applying the chain rule, we differentiate: \[ \frac{dy}{dx} = \frac{1}{\log(7)} \cdot \frac{1}{\sqrt{x^{2}-9}} \cdot \frac{d}{dx}\left(\sqrt{x^{2}-9}\right). \] Next, we differentiate \(\sqrt{x^{2}-9}\): \[ \frac{d}{dx}\left(\sqrt{x^{2}-9}\right) = \frac{1}{2\sqrt{x^{2}-9}} \cdot (2x) = \frac{x}{\sqrt{x^{2}-9}}. \] So, putting it all together: \[ \frac{dy}{dx} = \frac{1}{\log(7)} \cdot \frac{1}{\sqrt{x^{2}-9}} \cdot \frac{x}{\sqrt{x^{2}-9}} = \frac{x}{\log(7) \cdot (x^{2}-9)}. \] Thus, the derivative is: \( y'=\frac{x}{\log(7) \cdot (x^{2}-9)} \)