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).

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.