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The problem asks to find \( h'(1) \) given that \( h(x) = f(g(x)) \) and the table of values for \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \).

### Solution:
We can apply the **chain rule** to differentiate \( h(x) = f(g(x)) \):

\[
h'(x) = f'(g(x)) \cdot g'(x)
\]

Now, to find \( h'(1) \), we need the values of \( f'(g(1)) \) and \( g'(1) \) from the table.

From the table:
- \( g(1) = 1 \)
- \( g'(1) = 2 \)
- \( f'(1) = 2 \) (because \( g(1) = 1 \))

Now substitute these into the chain rule formula:

\[
h'(1) = f'(g(1)) \cdot g'(1) = f'(1) \cdot g'(1) = 2 \cdot 2 = 4
\]

Thus, \( h'(1) = 4 \).

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Would you like more details or have any other questions?

Here are 5 related questions you might find useful:
1. What is the chain rule and how is it used in calculus?
2. How would the derivative change if \( h(x) = f(x) \cdot g(x) \) instead?
3. What would happen if \( g'(x) \) were a constant function?
4. Can you explain why we evaluated \( f' \) at \( g(1) \) in this problem?
5. How do we handle derivatives when functions are composed more than once (e.g., \( f(g(h(x))) \))?

**Tip:** Always ensure you understand the chain rule, as it is essential for differentiating composite functions effectively.

If h(x) = f(g(x)), find h'(1) given the values of f(x), f'(x), g(x), and g'(x) at x = 1.

Learn how to find the derivative h'(1) for the composite function h(x) = f(g(x)) using the chain rule. Given values for f(x), f'(x), g(x), and g'(x), this problem demonstrates key calculus concepts on differentiation and composite functions. Suitable for Grades 11-12.

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