https://file.aiquickdraw.com/mathbot/file/1728093979126qt5x49ui.jpg

The function graphed above has:

Positive derivative on the interval(s) 
Incorrect

Negative derivative on the interval(s)

Let's analyze the graph to determine where the derivative of the function is positive or negative.

### **Positive Derivative**
The derivative of a function is positive when the graph is increasing, meaning that the curve is sloping upwards. From the graph, the function is increasing on the interval:
- Approximately from \( x = -3 \) to \( x = -1 \).

### **Negative Derivative**
The derivative is negative when the function is decreasing, or the curve is sloping downwards. Based on the graph:
- The function is decreasing on two intervals: from \( x = -5 \) to \( x = -3 \) and from \( x = -1 \) to \( x = 5 \).

### Summary:
- **Positive derivative**: \( (-3, -1) \).
- **Negative derivative**: \( (-5, -3) \cup (-1, 5) \).

Would you like more details, or have any questions?

Here are some related questions:
1. How do you calculate the derivative from a graph?
2. What does the second derivative tell us about the concavity of a function?
3. How can you identify critical points on this graph?
4. What are inflection points, and can we find any here?
5. How would the graph change if the function were shifted vertically?

**Tip**: The intervals of increasing or decreasing behavior correspond to where the first derivative is positive or negative, respectively.

The function graphed above has: Positive derivative on the interval(s). Negative derivative on the interval(s).

Learn how to analyze a graph and identify intervals where the derivative is positive (increasing) or negative (decreasing). This calculus problem involves understanding the first derivative test and graph interpretation. Suitable for Grades 10-12 students studying Calculus.

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