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.