To estimate $f'(9)$ based on the given table, we can use the average rate of change (slope) between the values of $f(x)$ that are close to $x = 9$. The two closest values of $x$ to 9 are 7 and 10.

We will use the slope formula for these two points to estimate the derivative:

$f'(9) \approx \frac{f(10) - f(7)}{10 - 7}$

From the table:

• $f(10) = -99$
• $f(7) = -89$

Now, calculate the slope:

$f'(9) \approx \frac{-99 - (-89)}{10 - 7} = \frac{-99 + 89}{3} = \frac{-10}{3} \approx -3.33$

So, the best estimate for $f'(9)$ is A. -3.33.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do we interpret the derivative $f'(x)$ in the context of this function?
2. What other methods could we use to estimate a derivative from a table of values?
3. How would the estimate change if we used points further from $x = 9$?
4. Why is the derivative important in understanding the behavior of a function?
5. How would the process differ if the function were not differentiable?

Tip: When estimating derivatives from a table, always choose points as close as possible to the desired $x$-value for greater accuracy.