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…

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)= )

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)= )

Answer

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 )