Calculate the derivative of the following function. \[ y=(8-\sqrt{x})^{9} \] \( \frac{d}{d x}=\square \)

To find the derivative of the function \( y=(8-\sqrt{x})^{9} \), we can use the chain rule. Let \( u = 8 - \sqrt{x} \), then \( y = u^9 \). The derivative \( y' \) can be found by first finding \( \frac{dy}{du} \) and then \( \frac{du}{dx} \). 1. Compute \( \frac{dy}{du} = 9u^8 \). 2. Now find \( \frac{du}{dx} = -\frac{1}{2\sqrt{x}} \) since the derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \). Now, we apply the chain rule: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 9u^8 \cdot \left(-\frac{1}{2\sqrt{x}}\right). \] Substituting \( u \) back in gives: \[ \frac{dy}{dx} = 9(8 - \sqrt{x})^8 \cdot \left(-\frac{1}{2\sqrt{x}}\right) = -\frac{9(8 - \sqrt{x})^8}{2\sqrt{x}}. \] Thus, the final answer is: \[ \frac{d}{dx}y = -\frac{9(8 - \sqrt{x})^8}{2\sqrt{x}}. \]