The question asks to find $f'(9)$ for the given function $f(x) = \frac{4}{\sqrt{x}}$.

Step 1: Express $f(x)$ in a simpler form

The function can be rewritten as: $f(x) = 4x^{-\frac{1}{2}}$

Step 2: Differentiate $f(x)$

Now, apply the power rule of differentiation: $f'(x) = 4 \times -\frac{1}{2} x^{-\frac{3}{2}} = -2x^{-\frac{3}{2}}$

Step 3: Find $f'(9)$

Substitute $x = 9$ into $f'(x)$: $f'(9) = -2 \times 9^{-\frac{3}{2}} = -2 \times \frac{1}{27} = -\frac{2}{27}$

Thus, $f'(9) = -\frac{2}{27}$, which corresponds to Answer A.

Would you like further explanation on the steps, or do you have any specific questions?

Here are 5 related questions to expand your understanding:

1. How do you apply the power rule of differentiation to more complex functions?
2. What is the general process for differentiating functions with fractional exponents?
3. How would you approach the derivative of functions involving roots of different orders?
4. Can you explain the difference between power functions and other types of functions?
5. What are the applications of this kind of differentiation in physics and engineering?

Tip: Always simplify the function before differentiating to make the process easier and more accurate.