Question

let ( f(x)=f(f(x)) ) and ( g(x)=(f(x))^{2} ) and suppose that ( f(8)=10, f(10)=2, f^{prime}(10)=15, f^{prime}(8)=12 ). find ( f^{prime}(8) ) and ( g^{prime}(8) ). ( f^{prime}(8)= ) ( g^{prime}(8)= )

Explanation:

Step1: Find \( F'(x) \) using the chain rule

The chain rule states that if \( F(x)=f(f(x)) \), then \( F'(x)=f'(f(x))\cdot f'(x) \).
Substitute \( x = 8 \): \( F'(8)=f'(f(8))\cdot f'(8) \).
Since \( f(8) = 10 \) and \( f'(8)=12 \), \( f'(f(8))=f'(10) = 15 \).
So \( F'(8)=15\times12 \).

Step2: Calculate \( F'(8) \)

\( F'(8)=15\times12 = 180 \).

Step3: Find \( G'(x) \) using the chain rule

If \( G(x)=(F(x))^{2} \), then by the chain rule \( G'(x)=2F(x)\cdot F'(x) \).
Substitute \( x = 8 \): \( G'(8)=2F(8)\cdot F'(8) \).
Since \( F(8)=f(f(8))=f(10) = 2 \) and \( F'(8)=180 \).
So \( G'(8)=2\times2\times180 \).

Step4: Calculate \( G'(8) \)

\( G'(8)=2\times2\times180=720 \).

Answer:

\( F'(8)=180 \), \( G'(8)=720 \)