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You are asked to compute two derivatives using the given table for the functions \( f(x) \), \( g(x) \), \( f'(x) \), and \( g'(x) \):

1. **\( h'(4) \)** where \( h(x) = f(g(x)) \)
2. **\( p'(3) \)** where \( p(x) = g(f(x)) \)

### Part a: \( h'(4) = f'(g(4)) \cdot g'(4) \)

By the chain rule, for \( h(x) = f(g(x)) \):
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
Substitute \( x = 4 \):
\[
h'(4) = f'(g(4)) \cdot g'(4)
\]

From the table:
- \( g(4) = 5 \)
- \( g'(4) = 2 \)
- \( f'(5) = 3 \)

Thus:
\[
h'(4) = f'(5) \cdot g'(4) = 3 \cdot 2 = 6
\]

### Part b: \( p'(3) = g'(f(3)) \cdot f'(3) \)

For \( p(x) = g(f(x)) \), again using the chain rule:
\[
p'(x) = g'(f(x)) \cdot f'(x)
\]
Substitute \( x = 3 \):
\[
p'(3) = g'(f(3)) \cdot f'(3)
\]

From the table:
- \( f(3) = 2 \)
- \( f'(3) = 3 \)
- \( g'(2) = 4 \)

Thus:
\[
p'(3) = g'(2) \cdot f'(3) = 4 \cdot 3 = 12
\]

### Final Answers:
- \( h'(4) = 6 \)
- \( p'(3) = 12 \)

Let me know if you need further details or explanations.

### 5 Related Questions:
1. How do you apply the chain rule to composite functions in general?
2. What would \( h'(2) \) be based on the given table?
3. Can you calculate \( p'(4) \) with the same method?
4. How is the chain rule derived from the definition of a derivative?
5. What is the product rule, and how is it different from the chain rule?

### Tip:
When dealing with derivatives of composite functions, always remember to apply the chain rule carefully—differentiate the outer function, 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'(4) b. p'(3).

Learn how to calculate the derivatives h'(4) and p'(3) for composite functions h(x) = f(g(x)) and p(x) = g(f(x)) using the chain rule. This problem utilizes a table of function values and their derivatives. Suitable for students in Grades 11-12 studying calculus.

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