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To estimate \( f'(5) \), we need to find the slope of the tangent line to the graph of \( f(x) \) at \( x = 5 \).

We can use the following process:

1. Identify points on the graph near \( x = 5 \). This will help approximate the slope using the difference quotient:
   \[
   f'(5) \approx \frac{f(x_2) - f(x_1)}{x_2 - x_1}
   \]
   where \( x_2 \) and \( x_1 \) are values near 5.

From the graph:
- At \( x = 5 \), \( f(5) \approx 7 \).
- Choosing another nearby point, at \( x = 4 \), we have \( f(4) \approx 5 \).

Now, we can estimate the slope:
\[
f'(5) \approx \frac{f(5) - f(4)}{5 - 4} = \frac{7 - 5}{1} = 2
\]

Thus, the estimated value of \( f'(5) \) is approximately **2**.

Would you like further details on this process or a deeper explanation?

Here are 5 related questions to explore this concept further:
1. How do you interpret the meaning of the derivative at a specific point in practical terms?
2. Can you explain why we use the difference quotient to approximate the derivative?
3. How would the approximation change if we used different points near \( x = 5 \)?
4. What is the significance of the tangent line to a graph at a particular point?
5. How do you calculate the exact derivative if you are given the function \( f(x) \) algebraically?

**Tip**: When estimating derivatives from a graph, choose points as close as possible to the target \( x \)-value for better accuracy.

Estimate f'(5) from the graph of y = f(x).

Learn how to estimate the derivative f'(5) from a graph of y = f(x). This problem involves finding the slope of the tangent line at a specific point using the difference quotient approximation.

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