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!

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!