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We are given a composite function \( r(x) = f(g(h(x))) \), and we are tasked with finding \( r'(1) \). The values provided are:

- \( h(1) = 3 \)
- \( g(3) = 2 \)
- \( h'(1) = 9 \)
- \( g'(3) = -2 \)
- \( f'(2) = 5 \)

### Step-by-Step Solution

We can use the chain rule for derivatives. For the composite function \( r(x) = f(g(h(x))) \), the derivative \( r'(x) \) is calculated as:

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

Now, let's evaluate this at \( x = 1 \):

1. **Find \( h(1) \):** We are given \( h(1) = 3 \), so \( h(1) = 3 \).
   
2. **Find \( g(h(1)) = g(3) \):** We are given \( g(3) = 2 \), so \( g(h(1)) = g(3) = 2 \).
   
3. **Find \( f'(g(h(1))) = f'(2) \):** We are given \( f'(2) = 5 \), so \( f'(g(h(1))) = f'(2) = 5 \).
   
4. **Find \( g'(h(1)) = g'(3) \):** We are given \( g'(3) = -2 \), so \( g'(h(1)) = g'(3) = -2 \).
   
5. **Find \( h'(1) \):** We are given \( h'(1) = 9 \), so \( h'(1) = 9 \).

### Now, put everything together:

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

Substitute the known values:

\[
r'(1) = 5 \cdot (-2) \cdot 9
\]

Simplifying:

\[
r'(1) = 5 \cdot (-18) = -90
\]

Thus, \( r'(1) = -90 \).

Let me know if you would like further details or have any questions!

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Here are 5 related questions to expand your understanding:

1. What is the chain rule and how is it applied to composite functions?
2. How would you differentiate a composite function if more than three nested functions were involved?
3. If \( f(x) \), \( g(x) \), and \( h(x) \) were trigonometric functions, how would the process change?
4. How do you interpret the physical meaning of a derivative in such composite functions?
5. How would the result change if one of the derivative values was zero?

**Tip:** Always write down each step of substitution clearly when working with chain rule problems—it helps avoid mistakes with nested derivatives.

Let r(x) = f(g(h(x))), where h(1) = 3, g(3) = 2, h'(1) = 9, g'(3) = -2, and f'(2) = 5. Find r'(1).

Learn how to apply the chain rule to differentiate a composite function, using given values for h(1), g(3), h'(1), g'(3), and f'(2) to find r'(1). This problem is suitable for Grade 12 or early college-level students.

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