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!