To solve for $g'(2)$ where $g(x) = f^{-1}(x)$, we can use the inverse function derivative formula:

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

In this case, we need to find $g'(2)$, which means we want the derivative of the inverse function at $x = 2$.

1. Step 1: Find $g(2)$
   Since $g(x) = f^{-1}(x)$, we are looking for an $a$ such that $f(a) = 2$.

   From the table:

   • When $x = 1$, $f(x) = 2$.

   Therefore, $g(2) = 1$.

2. Step 2: Calculate $g'(2)$ using $f'(g(2))$
   We now substitute $g(2) = 1$ into the derivative formula:

   $g'(2) = \frac{1}{f'(g(2))} = \frac{1}{f'(1)}$

   From the table, $f'(1) = -8$. So,

   $g'(2) = \frac{1}{-8} = -\frac{1}{8}$

Final Answer:

$g'(2) = -\frac{1}{8}$

Would you like further explanation on any step?

---

Related Questions:

1. How do we interpret the inverse function derivative formula geometrically?
2. What happens to $g'(x)$ when $f'(x) = 0$?
3. How can we approximate $g(x)$ values when $f(x)$ data points are limited?
4. How would $g'(x)$ change if $f(x)$ were decreasing?
5. Can the inverse function theorem apply to functions that are not differentiable?

Tip: When finding derivatives of inverse functions, identifying points on the original function that match the given criteria simplifies calculations significantly.