Question

Let F(x) = f(x ^ 7) and G(x) = (f(x)) ^ 7 You also know that , f(a) = 2 , f^ prime (a)=2 , f^ prime (a^ 7 )=13; a ^ 6 = 8

Answer 1

Given

`\begin{gathered} F(x)=f(x^7) \\ G(x)=(f(x))^7 \end{gathered}`

And,

`a^6=8,\text{ }f(a)=2,\text{ }f^(\prime)(a)=2,\text{ }f^(\prime)(a^7)=13`

To find:

The value of F'(a) and G'(a).

Step-by-step explanation:

It is given that,

`\begin{gathered} F(x)=f(x^7) \\ G(x)=(f(x))^7 \end{gathered}`

And,

`a^6=8,\text{ }f(a)=2,\text{ }f^(\prime)(a)=2,\text{ }f^(\prime)(a^7)=13`

That implies,

`\begin{gathered} F^(\prime)(x)=(d)/(dx)(f(x^7)) \\ =f^(\prime)(x^7)\cdot7x^6 \\ F^(\prime)(a)=f^(\prime)(a^7)\cdot7a^6 \\ =13\cdot(7*8) \\ =13*56 \\ =728 \end{gathered}`

Also,

`\begin{gathered} G^(\prime)(x)=(d)/(dx)((f(x))^7) \\ =7(f(x))^6\cdot f^(\prime)(x) \\ G^(\prime)(a)=7(f(a))^6\cdot f^(\prime)(a) \\ =7*(2)^6\cdot2 \\ =7*2^7 \\ =7*128 \\ =896 \end{gathered}`

Hence, the value of F'(a)=728, and the value of G'(a)=896.