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The problem asks to find the derivative of the function \( f(x) = \sqrt{x^3} + \frac{\sqrt{x}}{2} \) and evaluate it at \( x = 3 \), expressing the result as a single fraction in simplest radical form.

### Step-by-Step Solution:

1. **Rewrite the function for easier differentiation:**
   \[
   f(x) = x^{3/2} + \frac{1}{2} x^{1/2}
   \]

2. **Differentiate term by term:**
   - The derivative of \( x^{3/2} \) is:
     \[
     \frac{d}{dx} \left( x^{3/2} \right) = \frac{3}{2} x^{1/2}
     \]
   - The derivative of \( \frac{1}{2} x^{1/2} \) is:
     \[
     \frac{d}{dx} \left( \frac{1}{2} x^{1/2} \right) = \frac{1}{4} x^{-1/2}
     \]

3. **Combine the results:**
   \[
   f'(x) = \frac{3}{2} x^{1/2} + \frac{1}{4} x^{-1/2}
   \]

4. **Evaluate at \( x = 3 \):**
   - \( x^{1/2} = \sqrt{3} \)
   - \( x^{-1/2} = \frac{1}{\sqrt{3}} \)

   Substituting into the derivative:
   \[
   f'(3) = \frac{3}{2} \sqrt{3} + \frac{1}{4} \cdot \frac{1}{\sqrt{3}} = \frac{3 \sqrt{3}}{2} + \frac{1}{4 \sqrt{3}}
   \]

5. **Simplify the expression:**
   To combine the two terms, express them with a common denominator:
   \[
   \frac{3 \sqrt{3}}{2} + \frac{1}{4 \sqrt{3}} = \frac{6 \sqrt{3}}{4} + \frac{1}{4 \sqrt{3}} = \frac{6 \sqrt{3} + 1}{4 \sqrt{3}}
   \]

6. **Rationalize the denominator:**
   Multiply both the numerator and denominator by \( \sqrt{3} \):
   \[
   \frac{6 \sqrt{3} + 1}{4 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{(6 \sqrt{3} + 1) \cdot \sqrt{3}}{4 \cdot 3} = \frac{18 + \sqrt{3}}{12}
   \]

Thus, the simplified answer is:
\[
f'(3) = \frac{18 + \sqrt{3}}{12}
\]

Let me know if you want more details or if you have any questions!

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Here are 5 related questions to deepen understanding:

1. How do you differentiate functions that include square roots and fractional exponents?
2. What is the process for rationalizing a denominator, and why is it useful?
3. How would you solve for the derivative of a function that includes higher powers of \( x \) under a square root?
4. What are other techniques for simplifying expressions involving radicals and fractions?
5. How can the chain rule be applied to more complex functions involving nested radicals?

**Tip:** When working with square roots in calculus, rewrite them using fractional exponents to make differentiation easier!

Given the function f(x) = √(x³) + √(x)/2, find f'(3). Express your answer as a single fraction in simplest radical form.

Learn how to differentiate the function f(x) = √(x³) + √(x)/2 and evaluate the derivative at x = 3. The solution is expressed as a single fraction in simplest radical form, covering key calculus concepts like the power rule and rationalizing denominators. Suitable for Grade 12 or early college students.

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