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To solve this problem, we need to find the derivative \( g'(4) \), where \( g(x) = \sqrt{f(x)} \) and \( f(x) \) is the function represented by the red curve on the graph. The problem states that there is a tangent line to \( f(x) \) at \( x = 4 \), shown as the blue line.

### Solution Steps

1. **Identify the relationship between \( g(x) \) and \( f(x) \):**
   Since \( g(x) = \sqrt{f(x)} \), we can use the chain rule to find \( g'(x) \) in terms of \( f(x) \) and \( f'(x) \):
   \[
   g'(x) = \frac{d}{dx} \left( \sqrt{f(x)} \right) = \frac{1}{2\sqrt{f(x)}} \cdot f'(x)
   \]

2. **Evaluate \( f(4) \):**
   From the graph, find the value of \( f(4) \), which is the \( y \)-coordinate of \( f(x) \) when \( x = 4 \). According to the graph, \( f(4) \approx 4 \).

3. **Evaluate \( f'(4) \):**
   The derivative \( f'(4) \) represents the slope of the tangent line to \( f(x) \) at \( x = 4 \). Based on the graph, we can estimate the slope of this tangent line (blue line). By observing the slope, it appears to go from approximately \( y = 4 \) at \( x = 4 \) to \( y = 0 \) at \( x = 6 \), giving us:
   \[
   f'(4) \approx \frac{0 - 4}{6 - 4} = \frac{-4}{2} = -2
   \]

4. **Calculate \( g'(4) \):**
   Substitute \( f(4) = 4 \) and \( f'(4) = -2 \) into the expression for \( g'(x) \):
   \[
   g'(4) = \frac{1}{2\sqrt{4}} \cdot (-2) = \frac{1}{2 \cdot 2} \cdot (-2) = \frac{1}{4} \cdot (-2) = -\frac{1}{2}
   \]

### Final Answer
\[
g'(4) = -\frac{1}{2}
\]

Would you like further explanation on any part of this solution? 

Here are some related questions to deepen understanding:
1. What is the chain rule, and how is it applied in this example?
2. How can we estimate the slope of a tangent line more precisely?
3. Why is it necessary to evaluate both \( f(4) \) and \( f'(4) \) to find \( g'(4) \)?
4. What other methods can be used to approximate the derivative at a point on a graph?
5. How would \( g(x) \) change if \( g(x) = f(x)^2 \) instead?

**Tip:** When working with graphs, use points on the tangent line to calculate the slope accurately by choosing coordinates that are easy to identify.

The graph of f(x) and a line tangent to the curve at x = 4 are shown. If g(x) = √f(x), evaluate g'(4).

This solution demonstrates how to find the derivative g'(4) when g(x) = √f(x) and f(x) is given by a graph. Using the Chain Rule and tangent line slope estimation, this problem is ideal for high school calculus students (Grades 11-12).

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