Let 
r(x) = f(g(h(x))),
 where 
h(1) = 3, g(3) = 4, h'(1) = 5, g'(3) = 5,
 and 
f '(4) = 8.
 Find 
r'(1).

We are given the function \( r(x) = f(g(h(x))) \), and we need to find \( r'(1) \).

We can apply the **chain rule** for composition of functions. The chain rule states that if \( r(x) = f(g(h(x))) \), then

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

### Step 1: Differentiate using the chain rule
Using the chain rule, we have:

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

### Step 2: Evaluate \( r'(1) \)
We need to evaluate each part of this expression at \( x = 1 \).

#### 1. Evaluate \( h(1) \)
We are given that:
\[
h(1) = 3
\]

#### 2. Evaluate \( g(3) \)
Since \( h(1) = 3 \), we substitute into \( g(h(1)) = g(3) \). We are also given that:
\[
g(3) = 4
\]

#### 3. Evaluate \( f'(4) \)
Now, we need \( f'(g(h(1))) = f'(4) \). We are given:
\[
f'(4) = 8
\]

#### 4. Evaluate \( g'(3) \)
Next, we use \( g'(h(1)) = g'(3) \). We are given:
\[
g'(3) = 5
\]

#### 5. Evaluate \( h'(1) \)
Finally, we are given that:
\[
h'(1) = 5
\]

### Step 3: Substitute and calculate
Now we can substitute everything into the chain rule expression for \( r'(1) \):

\[
r'(1) = f'(g(h(1))) \cdot g'(h(1)) \cdot h'(1)
\]
\[
r'(1) = f'(4) \cdot g'(3) \cdot h'(1)
\]
\[
r'(1) = 8 \cdot 5 \cdot 5
\]

\[
r'(1) = 200
\]

Thus, the value of \( r'(1) \) is \( \boxed{200} \).

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this:
1. How does the chain rule apply to nested functions in general?
2. Can you explain the meaning of each derivative term in this specific example?
3. What happens if one of the derivative values is zero? How does it affect the result?
4. How would this process change if one of the functions was a product rather than a composition?
5. Can the chain rule be extended to functions with more than three compositions?

**Tip:** When applying the chain rule, always differentiate from the outermost function to the innermost, while keeping track of each individual derivative!

Let r(x) = f(g(h(x))), where h(1) = 3, g(3) = 4, h'(1) = 5, g'(3) = 5, and f '(4) = 8. Find r'(1).

Learn how to apply the chain rule to differentiate r(x) = f(g(h(x))) with given values for h(1), g(3), and their derivatives. This step-by-step solution explores key calculus concepts and shows how to compute r'(1) using the chain rule. Suitable for undergraduate calculus students.

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