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The problem you uploaded asks to find the derivative of the function:

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

and evaluate it at \( x = 9 \).

### Step-by-step solution:

1. **Rewrite the function** using logarithms:
   \[
   f(x) = x^{\sqrt{x}} = e^{\ln(x^{\sqrt{x}})} = e^{\sqrt{x} \ln(x)}
   \]

   So now we have \( f(x) = e^{\sqrt{x} \ln(x)} \).

2. **Differentiate using the chain rule**:
   Let \( u(x) = \sqrt{x} \ln(x) \), then \( f(x) = e^{u(x)} \).
   
   The derivative of \( f(x) \) is:
   \[
   f'(x) = e^{u(x)} \cdot u'(x)
   \]
   
   Now, let's differentiate \( u(x) \):
   \[
   u(x) = \sqrt{x} \ln(x) = x^{1/2} \ln(x)
   \]
   
   To differentiate \( u(x) \), use the product rule:
   \[
   u'(x) = \frac{d}{dx}\left( x^{1/2} \ln(x) \right) = \left( \frac{1}{2}x^{-1/2} \ln(x) \right) + \left( x^{1/2} \cdot \frac{1}{x} \right)
   \]
   Simplifying:
   \[
   u'(x) = \frac{1}{2\sqrt{x}} \ln(x) + \frac{1}{\sqrt{x}}
   \]

3. **Substitute back into the derivative of \( f(x) \)**:
   \[
   f'(x) = e^{\sqrt{x} \ln(x)} \cdot \left( \frac{1}{2\sqrt{x}} \ln(x) + \frac{1}{\sqrt{x}} \right)
   \]

4. **Evaluate at \( x = 9 \)**:
   First, calculate the components at \( x = 9 \):
   \[
   \sqrt{9} = 3, \quad \ln(9) = 2 \ln(3)
   \]
   So:
   \[
   u(9) = 3 \cdot 2 \ln(3) = 6 \ln(3)
   \]
   \[
   e^{u(9)} = e^{6 \ln(3)} = 3^6 = 729
   \]
   Now, calculate \( u'(9) \):
   \[
   u'(9) = \frac{1}{6} \cdot 2 \ln(3) + \frac{1}{3} = \frac{\ln(3)}{3} + \frac{1}{3}
   \]
   Therefore:
   \[
   f'(9) = 729 \left( \frac{\ln(3)}{3} + \frac{1}{3} \right) = 729 \cdot \frac{\ln(3) + 1}{3}
   \]
   Simplify:
   \[
   f'(9) = 243 (\ln(3) + 1)
   \]

### Final answer:
\[
f'(9) = 243 (\ln(3) + 1)
\]

Would you like any further details or clarifications?

### Follow-up questions:
1. How does the product rule apply in this situation, and how does it simplify the derivative?
2. What is the chain rule, and why is it used when differentiating \( e^{\sqrt{x} \ln(x)} \)?
3. Can you explain the logarithmic property \( \ln(x^{a}) = a \ln(x) \) and how it helps simplify the function?
4. How can we generalize this approach for other exponentials of the form \( x^{g(x)} \)?
5. How does knowing the natural logarithm of common numbers like \( \ln(3) \) aid in solving problems like this?

### Tip:
Whenever you encounter functions with exponents involving variables, applying logarithmic differentiation often simplifies the process of finding the derivative.

Learn how to find the derivative of the function f(x) = x^√x at x=9 using logarithmic differentiation, the chain rule, and the product rule. Step-by-step solution ideal for college-level calculus students.

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