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!