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The problem asks for the estimation of the derivative of the function \( f(x) \) at \( x = -1 \) by examining the graph.

To estimate the derivative \( f'(-1) \), we need to calculate the slope of the tangent line to the curve at \( x = -1 \).

From the graph:
- At \( x = -1 \), the function \( f(x) \) is around \( y = 2 \).
- We can estimate the slope of the tangent line by observing how \( f(x) \) changes around \( x = -1 \). The function is increasing as \( x \) moves from left to right at this point.

To estimate the slope:
1. Between \( x = -2 \) and \( x = 0 \), the function rises from approximately \( y = -1 \) at \( x = -2 \) to \( y = 4 \) at \( x = 0 \).
   
   Slope between these points:
   \[
   \text{slope} = \frac{\Delta y}{\Delta x} = \frac{4 - (-1)}{0 - (-2)} = \frac{5}{2} = 2.5
   \]
   
   Since the graph appears somewhat linear near \( x = -1 \), the derivative \( f'(-1) \) is likely close to 2.5.

Thus, the estimated value of the derivative at \( x = -1 \) is approximately **2.5**.

Would you like to explore more details about estimating derivatives from graphs?

Here are 5 related questions to extend the learning:
1. How do you determine the slope of a curve at a specific point?
2. What is the significance of a derivative in relation to a graph?
3. How can you visually distinguish between increasing and decreasing slopes on a graph?
4. How would the derivative behave at a local maximum or minimum?
5. What are the steps to compute the derivative algebraically if the function was provided?

**Tip:** The steeper the graph, the higher the derivative value. Conversely, if the graph flattens, the derivative approaches zero.

Consider the graph of y = f(x) shown below in blue. Estimate the value of the derivative at x = -1. Then select the answer that most closely matches your computed value for f'(-1).

Learn how to estimate the derivative f'(-1) from a graph. This problem covers concepts like slope, rate of change, and interpreting graphs to find the derivative. Suitable for students in Grades 11-12.

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