Let F(x) = f(x) and G(x) = (f(x)) and suppose that

a = 5, .

F(a) - 2, f'(a) = 6, f(a) = 13

Find F'(a) and G'(a).

F'(a) =

G' (a) =

Given the problem, we are tasked with finding \( F'(a) \) and \( G'(a) \) based on the following information:

- \( F(x) = f(x) \)
- \( G(x) = f(f(x)) \)
- \( a = 5 \)
- \( F(a) = f(a) = 13 \)
- \( F(a) - 2 = f(a) - 2 = 11 \)
- \( f'(a) = 6 \)
- \( f(a) = 13 \)

### 1. Finding \( F'(a) \):

Since \( F(x) = f(x) \), we have:
\[
F'(x) = f'(x)
\]
Therefore:
\[
F'(a) = f'(a) = 6
\]

### 2. Finding \( G'(a) \):

Given that \( G(x) = f(f(x)) \), we will use the chain rule to differentiate \( G(x) \):
\[
G'(x) = f'(f(x)) \cdot f'(x)
\]
Substituting \( a = 5 \) into the equation:
\[
G'(a) = f'(f(a)) \cdot f'(a)
\]

From the problem, we know that:
- \( f(a) = 13 \)
- \( f'(a) = 6 \)

We need to find \( f'(f(a)) \), which is \( f'(13) \). However, since \( f'(13) \) is not provided in the problem, we assume it remains symbolic:
\[
G'(a) = f'(13) \cdot 6
\]

Thus, \( G'(a) = 6f'(13) \).

### Final Answers:
- \( F'(a) = 6 \)
- \( G'(a) = 6 \cdot f'(13) \)

If you want to go further into this or have any questions, feel free to ask!

Here are 5 related questions:
1. How does the chain rule apply when differentiating composite functions?
2. If \( f'(13) \) were given, how would it affect the result for \( G'(a) \)?
3. Can you describe the significance of the derivative \( f'(x) \) in practical terms?
4. How would the result change if \( G(x) \) was defined differently, say \( G(x) = f(x^2) \)?
5. How can we interpret the physical meaning of \( G'(a) \) in an application context?

**Tip**: Remember that composite functions often require the chain rule, so always consider how functions are nested when differentiating!

Let F(x) = f(x) and G(x) = f(f(x)) and suppose that a = 5, F(a) = f(a) = 13, f'(a) = 6. Find F'(a) and G'(a).

Solve the differentiation problem involving F(x) = f(x) and G(x) = f(f(x)) with a detailed step-by-step solution using the chain rule. Given that a = 5, f(a) = 13, and f'(a) = 6, learn how to find F'(a) and G'(a) symbolically. Suitable for advanced high school or college-level calculus students.

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