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The image contains a worksheet related to studying the variations of a function and its derivative using the software **GeoGebra**. The exercise focuses on the visualization of the derivative of the quadratic function \( f(x) = x^2 \). Here's an outline of the key parts:

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### **Instructions**:
1. **Launch GeoGebra** and enter the function \( f(x) = x^2 \).
2. Place a point \( A \) on the curve \( f \).
3. Trace the tangent to the curve at \( A \), and call this tangent \( T \), then change its color to red.
4. Define the slope of the tangent (derivative) and link it to point \( M \) with the same abscissa as \( A \) but with an ordinate equal to the slope.
5. Display point \( M \) in purple and activate its trace.
6. Make point \( A \) evolve along the curve.
7. Trace the derivative \( f' \) by entering the derivative formula \( f'(x) \).
8. Save your work under "dérivée fonction carrée."

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### **Key Questions**:
1. The derivative of \( f(x) = x^2 \) is **\( f'(x) = 2x \)**.
2. When graphing, the derivative is a **line**.

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### **Exercise**:
In the table provided:
- When the function is **increasing**, the derivative is **positive**.
- When the function is **decreasing**, the derivative is **negative**.

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### **Conclusion**:
Knowing the sign of the derivative allows understanding the **variations of the function**.

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Let me know if you'd like further explanation or detailed steps for using GeoGebra in this context.

### 5 Follow-up Questions:
1. What does the slope of the tangent line represent in relation to the function?
2. How do we interpret the points where the derivative is zero?
3. How can we use the derivative to identify the function's increasing or decreasing intervals?
4. What happens to the tangent when \( A \) is moved along the curve?
5. How does the trace of point \( M \) reflect the changes in the function's slope?

**Tip**: When working with derivatives, always check the behavior at critical points where the derivative equals zero!

Étudier les variations d'une fonction et sa dérivée avec GeoGebra pour la fonction f(x) = x^2.

Learn how to study the variations of a function and its derivative with GeoGebra, focusing on the quadratic function f(x) = x^2. This activity explores how the derivative helps identify the increasing and decreasing intervals of the function. Suitable for high school students in Grades 10-12.

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