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The given function is:

\[
f(x) = x \sqrt{x}
\]

### Step 1: Rewrite the function in a more convenient form for differentiation.

\[
f(x) = x \cdot x^{1/2} = x^{3/2}
\]

### Step 2: Differentiate the function using the power rule.

The power rule states that if \( f(x) = x^n \), then \( f'(x) = n x^{n-1} \).

Applying the power rule to \( f(x) = x^{3/2} \):

\[
f'(x) = \frac{3}{2} x^{\frac{3}{2} - 1} = \frac{3}{2} x^{1/2}
\]

### Step 3: Evaluate \( f'(9) \).

Substitute \( x = 9 \) into \( f'(x) \):

\[
f'(9) = \frac{3}{2} \cdot 9^{1/2} = \frac{3}{2} \cdot 3 = \frac{9}{2}
\]

So, the derivative evaluated at \( x = 9 \) is:

\[
f'(9) = \frac{9}{2}
\]

Would you like further details on this, or any questions?

Here are five related questions for deeper understanding:
1. How would the derivative change if the function were \( f(x) = x^2 \sqrt{x} \)?
2. What are the steps to find the second derivative \( f''(x) \) for this function?
3. Can you explain why we use the power rule in this situation?
4. What is the geometric interpretation of \( f'(9) \) in terms of the graph of \( f(x) \)?
5. How would you apply implicit differentiation to a function involving square roots?

**Tip:** The power rule is one of the most useful differentiation tools; make sure you recognize when to convert roots into fractional exponents for easier differentiation!

Learn how to differentiate the function f(x) = x√x using the power rule and evaluate the derivative at x = 9. This problem involves key calculus concepts such as the power rule for differentiation and is suitable for students in Grades 11-12.

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