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

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To find the derivative of the function $$ y = (8 - \sqrt{x})^9 $$, we will use the chain rule. The chain rule states that if you have a composite function $$ y = f(g(x)) $$, then the derivative is given by:

$$
\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
$$

In this case, let $$ u = 8 - \sqrt{x} $$, so that $$ y = u^9 $$.

First, we need to find the derivative of $$ y $$ with respect to $$ u $$:

$$
\frac{dy}{du} = 9u^8
$$

Next, we need to find the derivative of $$ u $$ with respect to $$ x $$:

$$
u = 8 - \sqrt{x} \implies \frac{du}{dx} = -\frac{1}{2\sqrt{x}}
$$

Now, we can 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 back $$ u = 8 - \sqrt{x} $$:

$$
\frac{dy}{dx} = 9(8 - \sqrt{x})^8 \cdot \left(-\frac{1}{2\sqrt{x}}\right)
$$

This simplifies to:

$$
\frac{dy}{dx} = -\frac{9(8 - \sqrt{x})^8}{2\sqrt{x}}
$$

Thus, the derivative of the function $$ y = (8 - \sqrt{x})^9 $$ is:

$$
\frac{dy}{dx} = -\frac{9(8 - \sqrt{x})^8}{2\sqrt{x}}
$$
The derivative of the function $$ y = (8 - \sqrt{x})^9 $$ is $$ \frac{dy}{dx} = -\frac{9(8 - \sqrt{x})^8}{2\sqrt{x}} $$.

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

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