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To solve this problem, we will need to use the chain rule to compute the derivatives of both functions \( h(x) = f(g(x)) \) and \( p(x) = g(f(x)) \), using the data from the table.

### a. \( h'(3) \) where \( h(x) = f(g(x)) \)
Using the chain rule:
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
Now, we need to calculate \( h'(3) \). From the table:
- \( g(3) = 1 \)
- \( f'(1) = -8 \) (since \( g(3) = 1 \), we look at the value of \( f'(1) \))
- \( g'(3) = \frac{1}{5} \)

Thus:
\[
h'(3) = f'(g(3)) \cdot g'(3) = f'(1) \cdot g'(3) = -8 \cdot \frac{1}{5} = -\frac{8}{5}
\]

### b. \( p'(2) \) where \( p(x) = g(f(x)) \)
Using the chain rule:
\[
p'(x) = g'(f(x)) \cdot f'(x)
\]
Now, we need to calculate \( p'(2) \). From the table:
- \( f(2) = 4 \)
- \( g'(4) = \frac{3}{5} \) (since \( f(2) = 4 \), we look at the value of \( g'(4) \))
- \( f'(2) = -6 \)

Thus:
\[
p'(2) = g'(f(2)) \cdot f'(2) = g'(4) \cdot f'(2) = \frac{3}{5} \cdot (-6) = -\frac{18}{5}
\]

### Final Answers:
a. \( h'(3) = -\frac{8}{5} \)

b. \( p'(2) = -\frac{18}{5} \)

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Would you like more details on these steps or need further clarification?

Here are some related questions to further your understanding:

1. How is the chain rule applied when differentiating composite functions like \( f(g(x)) \)?
2. What would \( h'(2) \) be using the same method from the table?
3. How does the derivative of \( g(f(x)) \) change when the order of composition is reversed?
4. What is the significance of knowing both \( f'(x) \) and \( g'(x) \) in these problems?
5. Could you calculate the second derivative \( h''(x) \) for \( h(x) = f(g(x)) \) using the chain rule?

**Tip:** When using the chain rule, always differentiate the outer function first, then multiply by the derivative of the inner function.

Let h(x) = f(g(x)) and p(x) = g(f(x)). Use the table to compute the following derivatives: a. h'(3), b. p'(2).

Learn how to compute derivatives of composite functions h(x) = f(g(x)) and p(x) = g(f(x)) using the chain rule. This problem uses table data to find h'(3) and p'(2). Suitable for students in Grades 11-12 studying calculus.

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