To determine $g'(2)$ for $g(x) = f^{-1}(x)$, we need to use the relationship between the derivatives of inverse functions. For an inverse function $g(x) = f^{-1}(x)$, the derivative is given by:

$g'(x) = \frac{1}{f'(g(x))}$

We are asked to find $g'(2)$, so we need to evaluate $g'(2)$ using this formula.

Step-by-Step Solution:

1. Since $g(x) = f^{-1}(x)$, we have $g(2) = y$ where $f(y) = 2$.
2. From the table, $f(2) = 5$, so we need to find $y$ such that $f(y) = 2$.
3. Observing the table, $f(2) = 5$ and $f(4) = 2$, so $g(2) = 4$.
4. Now, substitute $g(2) = 4$ into the derivative formula: $g'(2) = \frac{1}{f'(g(2))} = \frac{1}{f'(4)}$
5. From the table, $f'(4) = -5$, so: $g'(2) = \frac{1}{-5} = -\frac{1}{5}$

Conclusion:

The correct answer is (C) $g'(2) = -\frac{1}{5}$.

Would you like more details or have any questions?

Here are five related questions:

1. How does the derivative formula for inverse functions work in general?
2. Why is it essential to use $f'(g(x))$ instead of $f'(x)$ when calculating $g'(x)$?
3. How can you verify the values of $g(x)$ in such inverse function problems?
4. What would $g'(0)$ be for this function if needed?
5. How does the behavior of $f(x)$ being decreasing affect $g'(x)$?

Tip: When dealing with inverse functions, remember to match the input and output of each function carefully, as it’s easy to mix them up!