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We are given the function:

\[
f(x) = \int_2^x g(t) dt
\]

and the graph of \( f(x) \). We need to determine \( g(x) \), which is the derivative:

\[
g(x) = f'(x)
\]

### Step 1: Analyze the Graph of \( f(x) \)
The given graph of \( f(x) \) is a straight line passing through the origin with a positive slope. This suggests that \( f(x) \) is a linear function.

A linear function has a constant derivative. Since \( g(x) = f'(x) \), we conclude that \( g(x) \) must be a constant function.

### Step 2: Identify the Correct Graph of \( g(x) \)
Among the given options, the one that represents a constant function (a horizontal line) is the correct choice.

From the given choices, **option (A) shows a constant function**, which matches our result.

### Conclusion:
The correct answer is **option (A)**.

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Would you like a more detailed explanation? Here are some related questions:

1. How do we apply the Fundamental Theorem of Calculus to find \( g(x) \)?
2. What does the slope of \( f(x) \) indicate about \( g(x) \)?
3. Can \( f(x) \) be nonlinear in this case? Why or why not?
4. How would the graph of \( g(x) \) look if \( f(x) \) were a quadratic function?
5. What happens if \( f(x) \) has a point of non-differentiability?

**Tip:** The derivative of a linear function is always a constant.

Given the graph of f(x) and the integral relationship f(x) = ∫[2 to x] g(t) dt, find y = g(x). Choose the correct graph for g(x) from the options (A, B, C, D, E).

Learn how to find the graph of g(x) from the graph of f(x), given that f(x) is defined as an integral involving g(t). Apply the Fundamental Theorem of Calculus to determine g(x) as the derivative of f(x). Suitable for students in Grades 11-12.

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