https://file.aiquickdraw.com/mathbot/file/1725465655306r7kxn26b.jpg

The image shows a graph of a differentiable function \( f \) along with a tangent line to the graph of \( f \) at \( x = 2 \). The question asks for the value of \( f'(2) \).

To find \( f'(2) \), which represents the slope of the tangent line to the graph of \( f(x) \) at \( x = 2 \), we can analyze the slope of the tangent line shown in the graph.

### Steps to Solve:

1. **Identify the coordinates of points on the tangent line:**
   - From the graph, it appears that the tangent line passes through the points \((2, 2)\) and \((4, 4)\).

2. **Calculate the slope:**
   - The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
     \[
     m = \frac{y_2 - y_1}{x_2 - x_1}
     \]
   - Plugging in the coordinates of the two points:
     \[
     m = \frac{4 - 2}{4 - 2} = \frac{2}{2} = 1
     \]

3. **Conclusion:**
   - The slope of the tangent line at \( x = 2 \) is 1, so \( f'(2) = 1 \).

### Final Answer:
\( f'(2) = 1 \)

Would you like more details or have any further questions?

### Related Questions:
1. What does the derivative \( f'(x) \) represent in terms of the function \( f(x) \)?
2. How would you interpret the meaning of a negative derivative at a point?
3. How can you find the equation of the tangent line at \( x = 2 \)?
4. If the tangent line had a slope of 0, what would that imply about the function \( f(x) \) at \( x = 2 \)?
5. How would the calculation change if the function \( f(x) \) had a different behavior near \( x = 2 \)?

### Tip:
When analyzing graphs to determine slopes, try to use clear points on the line to minimize errors in calculating the slope.

Shown above is the graph of a differentiable function f along with the line tangent to the graph of f at x = 2. What is the value of f'(2)?

Discover how to find the value of f'(2) by analyzing the slope of the tangent line to a function's graph at x = 2. This problem focuses on key calculus concepts like differentiation and the geometric interpretation of the derivative. Perfect for students in Grades 11-12.

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