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Let's first tackle the differentiation of the given function and then the domain of the derivative.

### Step 1: Differentiate the function
The function is given as:

\[
f(x) = 7\sqrt{x - 8}
\]

We can rewrite this in exponent form for easier differentiation:

\[
f(x) = 7(x - 8)^{1/2}
\]

Now, we'll apply the chain rule to differentiate:

\[
f'(x) = 7 \cdot \frac{1}{2}(x - 8)^{-1/2} \cdot \frac{d}{dx}(x - 8)
\]

The derivative of \((x - 8)\) is 1, so we have:

\[
f'(x) = 7 \cdot \frac{1}{2}(x - 8)^{-1/2}
\]

Simplifying:

\[
f'(x) = \frac{7}{2\sqrt{x - 8}}
\]

### Step 2: Determine the domain of \( f'(x) \)
The domain of \( f'(x) \) is determined by ensuring the expression under the square root is positive, because square roots of negative numbers are not real. Therefore, we need:

\[
x - 8 > 0
\]

This implies:

\[
x > 8
\]

Thus, the domain of \( f'(x) \) is:

\[
x \in (8, \infty)
\]

### Final Answer:
1. The derivative is:

\[
f'(x) = \frac{7}{2\sqrt{x - 8}}
\]

2. The domain of \( f'(x) \) is:

\[
x \in (8, \infty)
\]

Would you like further clarification or more detailed explanations? 

Here are 5 related questions:
1. How do we apply the chain rule to functions more complex than \( \sqrt{x - 8} \)?
2. What happens if the argument of the square root function is negative?
3. How do you find the second derivative of this function?
4. What is the geometric interpretation of the derivative for a function like \( \sqrt{x} \)?
5. How would the domain change if we had \( f(x) = \sqrt{8 - x} \)?

**Tip:** Always verify the domain before differentiating functions involving radicals or fractions to avoid undefined expressions.

Differentiate the function f(x) = 7√(x - 8). Determine the domain of f'(x).

Learn how to differentiate the function f(x) = 7√(x - 8) and determine the domain of its derivative using the chain rule and radical function rules. Suitable for students in Grades 10-12.

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