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.