https://file.aiquickdraw.com/mathbot/file/1724740101204s6du5e7w.jpg

The problem in the image asks to find the values of \( f'(1) \) and \( f'(2) \) and then estimate \( f'(1.5) \) using these results. It involves the derivative \( f'(x) \) and possibly an interpolation technique.

### Let's solve the problem:

1. **Find \( f'(1) \):**
   The image likely provides a method or formula to compute \( f'(1) \). However, based on the structure of the problem, we assume that \( f'(1) \) is either provided directly or can be computed.

2. **Find \( f'(2) \):**
   Similarly, we should compute or extract \( f'(2) \) from the given data or formula.

3. **Estimate \( f'(1.5) \):**
   To estimate \( f'(1.5) \), we can use linear interpolation between \( f'(1) \) and \( f'(2) \).

The linear interpolation formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{(x_2 - x_1)}
\]
Here, \( x_1 = 1 \), \( y_1 = f'(1) \), \( x_2 = 2 \), \( y_2 = f'(2) \), and \( x = 1.5 \).

Once \( f'(1) \) and \( f'(2) \) are known, plug them into this formula to find \( f'(1.5) \).

### To proceed:
If the values of \( f'(1) \) and \( f'(2) \) are present in the problem statement and you'd like me to complete the calculation, please provide them. Alternatively, if you'd like me to make any other specific analysis or require the extraction of data from the image, let me know!

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### Related Questions:
1. How would you calculate \( f'(x) \) for different values of \( x \) if given a function \( f(x) \)?
2. What is linear interpolation, and why is it useful in estimating values between known points?
3. Can you explain the significance of derivatives in understanding the behavior of functions?
4. How would you use numerical methods to estimate derivatives if the function is complex?
5. What are the differences between forward, backward, and central difference methods in numerical differentiation?

**Tip:** When estimating derivatives at a non-integer point using known derivatives at integer points, linear interpolation is often a simple and effective method.

Learn how to estimate derivatives at non-integer points using linear interpolation. This calculus problem involves finding derivatives at specific points and using linear interpolation to estimate values between them. Suitable for advanced high school or college-level students.

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