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We are given that \( h(x) = f(g(x)) \), and we are tasked with finding \( h'(-5) \), with the following information:

- \( g(-5) = 9 \)
- \( g'(-5) = 2 \)
- \( f(9) = 12 \)
- \( f'(9) = 4 \)

### Step-by-step solution:

Since \( h(x) = f(g(x)) \), we need to use the chain rule to find \( h'(x) \). The chain rule states:

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

Now, substituting \( x = -5 \):

\[
h'(-5) = f'(g(-5)) \cdot g'(-5)
\]

From the given values:

- \( g(-5) = 9 \), so \( f'(g(-5)) = f'(9) \)
- \( g'(-5) = 2 \)
- \( f'(9) = 4 \)

Thus:

\[
h'(-5) = f'(9) \cdot g'(-5) = 4 \cdot 2 = 8
\]

### Final Answer:
\[
h'(-5) = 8
\]

Would you like more details on any part of the process? Here are five questions to expand your understanding:

1. How does the chain rule apply when differentiating composite functions like \( h(x) = f(g(x)) \)?
2. What would happen if the values of \( f'(9) \) or \( g'(-5) \) were different?
3. Can you explain why the derivative of \( f(g(x)) \) involves multiplying \( f'(g(x)) \) by \( g'(x) \)?
4. What are the conditions for applying the chain rule in general?
5. How would you solve this if the function involved more complex compositions (e.g., \( h(x) = f(g(h(x))) \))?

### Tip:
When dealing with composite functions, always check if the chain rule is required. It’s useful in breaking down the derivatives into manageable parts.

Given that h = f(g(x)), find h'(-5) if g(-5) = 9, g'(-5) = 2, f(9) = 12, and f'(9) = 4.

Learn how to use the chain rule to find h'(-5) when h(x) = f(g(x)) and given values for g(-5), g'(-5), f(9), and f'(9). This calculus problem demonstrates the application of differentiation techniques for composite functions, suitable for Grades 11-12 and college-level calculus students.

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