The problem asks to find the derivative of the function $f(x)$ at the point $x_0$, and the image shows a graph of the function $y = f(x)$ and its tangent line at $x_0$.

Key Observations:

1. Derivative Interpretation: The derivative of a function at a point $x_0$, denoted as $f'(x_0)$, is the slope of the tangent line to the graph of the function at that point.
2. Tangent Line Slope: From the graph, we can observe the green tangent line, and we need to determine its slope.

Determining the slope:

• The graph suggests the following:
  • At $x = 0$, the tangent line has a height of $y = 1$.
  • At $x = 1$, the tangent line has a height of $y = 2$.

Thus, over the interval from $x = 0$ to $x = 1$, the tangent line rises by 1 unit (from 1 to 2) and runs by 1 unit (from $x = 0$ to $x = 1$).

Therefore, the slope of the tangent line is: $f'(x_0) = \frac{\Delta y}{\Delta x} = \frac{2 - 1}{1 - 0} = 1$

Conclusion:

The value of the derivative $f'(x_0)$ is $\boxed{1}$.

Would you like further details or explanations on any step? Here are some related questions for you to consider:

1. How do you compute the derivative of a function algebraically?
2. What are the key characteristics of a tangent line to a curve?
3. What does the slope of a tangent line tell us about the behavior of a function?
4. How would the graph change if the derivative were negative?
5. What is the significance of the point where the tangent line meets the curve?

Tip: When interpreting graphs, always check the rise and run to find the slope accurately, especially when dealing with tangent lines.