We are tasked with finding $f'(3)$ given that $f(x) = g(h(x))$.

To solve this, we apply the chain rule:

$f'(x) = g'(h(x)) \cdot h'(x)$

From the table, when $x = 3$:

• $h(3) = 2$,
• $g'(h(3)) = g'(2)$, and $g'(2) = 0$ from the table,
• $h'(3) = 2$.

Using the chain rule: $f'(3) = g'(h(3)) \cdot h'(3) = g'(2) \cdot h'(3) = 0 \cdot 2 = 0$

Thus, $f'(3) = 0$.

Would you like further explanation, or do you have additional questions?

Here are 5 related questions to expand your understanding:

1. What is the chain rule, and how is it applied in general?
2. Can you find $f'(x)$ for a different value of $x$ using the same table?
3. Why does $g'(2) = 0$ cause $f'(3)$ to be zero, regardless of $h'(3)$?
4. How does the composition of functions affect their derivatives?
5. Could we extend this process to three or more nested functions?

Tip: Always identify intermediate values, like $h(x)$, before applying derivative rules for composite functions.