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To determine the derivative of \( f(x) \) at \( x = 0.4 \), we need to analyze the slope of the graph in the region that includes \( x = 0.4 \).

1. From the graph, we can see that from \( x = 0 \) to approximately \( x = 1 \), the function is increasing linearly. This means the derivative is constant in this region.
2. To find the derivative at \( x = 0.4 \), which lies in the linear increasing part, we calculate the slope of the line segment between \( x = 0 \) and \( x = 1 \).

### Step-by-step calculation:

- At \( x = 0 \), the value of \( f(x) \) is 0.
- At \( x = 1 \), the value of \( f(x) \) is 2.

The slope (which is the derivative for a linear segment) is given by the formula:

\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{f(1) - f(0)}{1 - 0} = \frac{2 - 0}{1 - 0} = 2
\]

Thus, the derivative of \( f(x) \) at \( x = 0.4 \) is **2**.

Would you like further details, or do you have any questions?

Here are five related questions you might find helpful:
1. How does the derivative change for different sections of the graph?
2. What is the meaning of the derivative in the context of this graph?
3. How can we determine the derivative at other points, such as \( x = 1.5 \)?
4. What is the slope of the graph when \( f(x) \) becomes constant?
5. How do we handle points where the graph has sharp corners or discontinuities?

**Tip:** The derivative represents the slope of the tangent line to the graph. For linear segments, the slope is constant.

Given the graph shown above of the function f(x), determine the derivative of f(x) at x = 0.4.

Learn how to determine the derivative of the function f(x) at x = 0.4 using graph interpretation. This calculus problem involves key concepts like derivatives and slopes. Suitable for students in Grades 11-12.

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