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The problem you've provided involves finding the derivative \( h'(-5) \) for the function \( h(x) = f(g(x)) \) using the given values:

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

We are tasked with using these values to find \( h'(-5) \). 

### Step-by-step solution:
Given \( h(x) = f(g(x)) \), we can apply the **chain rule** to differentiate \( h(x) \):

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

Now, let's substitute \( x = -5 \):

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

From the given information:
- \( g(-5) = 9 \), so \( h'(-5) = f'(9) \cdot g'(-5) \)
- \( f'(9) = 4 \)
- \( g'(-5) = 2 \)

Thus, we can now substitute these values:

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

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

---

Would you like more details or have any questions? Here are five related questions to expand your understanding:

1. How does the chain rule apply when differentiating composite functions?
2. What are the general steps for finding the derivative of a function like \( h(x) = f(g(x)) \)?
3. Why is knowing \( g(-5) \) and \( f'(9) \) important for this problem?
4. How do derivatives of composite functions change when the inner function is more complex?
5. How would you approach this if \( f'(9) \) or \( g'(-5) \) were different values?

### Tip:
Whenever you're working with composite functions, always identify the inner and outer functions clearly and apply the chain rule step-by-step.

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

Learn how to find the derivative h'(-5) for the composite function h(x) = f(g(x)) using the chain rule and given function values. This step-by-step guide explains key calculus concepts.

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