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The function provided is:

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

We are asked to compute the derivative \( f'(x) \), and specifically evaluate it at \( x = 5 \), \( x = 16 \), and \( x = -4 \), where the derivative exists.

### Step 1: Derivative of \( f(x) = \sqrt{x} \)

We can rewrite \( \sqrt{x} \) as \( x^{1/2} \). Using the power rule for differentiation:

\[
f'(x) = \frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}
\]

### Step 2: Evaluate the derivative at specific points

- **For \( x = 5 \):**

\[
f'(5) = \frac{1}{2\sqrt{5}} = \frac{1}{2 \times 2.236} \approx 0.2236
\]

- **For \( x = 16 \):**

\[
f'(16) = \frac{1}{2\sqrt{16}} = \frac{1}{2 \times 4} = \frac{1}{8} = 0.125
\]

- **For \( x = -4 \):**

Since the square root function \( f(x) = \sqrt{x} \) is not defined for negative values in the set of real numbers, the derivative does **not exist** for \( x = -4 \).

### Summary:
- \( f'(5) \approx 0.2236 \)
- \( f'(16) = 0.125 \)
- \( f'(-4) \) does not exist.

Would you like further clarification or more details on the steps? 

Here are 5 related questions:
1. How is the power rule used to differentiate functions like \( x^{n} \)?
2. Why is the derivative undefined for negative values under the square root?
3. How can we handle derivatives of more complex radical functions?
4. What are the rules for differentiating composite functions like \( f(g(x)) \)?
5. How can we use implicit differentiation for more advanced root-based functions?

**Tip**: Always check the domain of a function before differentiating to ensure the derivative exists at a given point!

Use technology to find f'(5), f'(16), and f'(-4) for the function f(x) = √x where the derivative exists.

Learn how to calculate the derivatives of f(x) = √x at points x = 5, x = 16, and x = -4. This solution includes the power rule and examines when the derivative is undefined for negative numbers under a square root. Suitable for students in Grades 11-12.

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