Question

How do you find the instantaneous rate of change from a table?

Answer 1

We approximate it by using the slope of the secant line through the two closest values to your target value.

Explanation:
Sometimes there are more requirement but in AP Calculus almost always end up taking the values to the left and right of your target value and calculating based on them.
$x|f\left(x\right)$
$1|3$
$2|5$
$3|4$
$4|9$

From the "table" we'd approximate the following:

$f^{\prime }\left(2\right)\approx \frac{f\left(3\right)-f\left(1\right)}{3-1}=\frac{1}{2}$

$f^{\prime }\left(3\right)\approx \frac{f\left(4\right)-f\left(2\right)}{4-2}=\frac{4}{2}=2$

For $1$ we'd have no choice but to do this:

$f^{\prime }\left(1\right)\approx \frac{f\left(2\right)-f\left(1\right)}{2-1}=\frac{2}{1}=2$

$f^{\prime }\left(4\right)\approx \frac{f\left(4\right)-f\left(3\right)}{4-3}=\frac{5}{1}=5$