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We are asked to find \( h'(3) \), where \( h(x) = f(g(x)) \), and we are provided a table with values for \( x \), \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \).

### Step-by-step approach:

1. Since \( h(x) = f(g(x)) \), we need to apply the chain rule to find \( h'(x) \):
   \[
   h'(x) = f'(g(x)) \cdot g'(x)
   \]
2. Now, to find \( h'(3) \), we need:
   - \( g(3) \)
   - \( f'(g(3)) \)
   - \( g'(3) \)

From the table:
- \( g(3) = 2 \) (looking at the row for \( g(x) \) where \( x = 3 \)).
- \( f'(2) = 3 \) (looking at the row for \( f'(x) \) where \( x = 2 \)).
- \( g'(3) = 4 \) (from the row for \( g'(x) \) where \( x = 3 \)).

Now, apply the chain rule:
\[
h'(3) = f'(g(3)) \cdot g'(3) = f'(2) \cdot g'(3) = 3 \cdot 4 = 12
\]

Thus, \( h'(3) = 12 \).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the chain rule and how does it work in general?
2. How would \( h'(x) \) change if \( h(x) = g(f(x)) \)?
3. Can we interpret \( h(x) = f(g(x)) \) as a composition of two functions, and how does this impact differentiation?
4. What does the derivative \( g'(x) \) represent in this context?
5. How would the result differ if \( f'(x) \) or \( g'(x) \) were negative?

**Tip**: Always double-check the table for the corresponding values, as small errors in reading values can affect the final result!

Find h'(3) if h(x) = f(g(x)) using the table of values provided.

Learn how to find the derivative h'(3) when h(x) = f(g(x)) using a table of values. This calculus problem involves the chain rule and understanding of function composition, suitable for high school students in Grades 11-12.

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