Let F(x)=f(x^9) and G(x)=(f(x))^9 and suppose that a^8=6, f(a)=2, f′(a)=12, f′(a^9)=4 find F'(a) and G'(a)

We begin by stating the given information: 

`F (x) = f(x^9)` ... Eq. 1.1

`G(x) = (f(x))^9` ... Eq. 1.2

Additionally we are given :

`a^8 = 6`  ... Eq. 2.1

`f(a) = 2` ... Eq. 2.2 

`f'(a) = 12` ... Eq. 2.3

`f'(a^9) = 4` ... Eq. 2.4

We are required to find: 

`F'(a) = ?`

And 

We have to use to above equation to solve what is required. Let's begin by solving F'(a): 

If `F(x) = f(x^9)`

` `

 Then using the chain rule :

`F'(x) = f'(x^9) * 9x^8`

Therefore applying ourselves, by substituting 'a' for 'x': 

`F' (a) = f'(a^9) * 9a^8`

We can now substitute Eq. 2.4 and Eq. 2.1

`F'(a) = 4 * 9(6) = 216` 

We have solved the first part, now let's solve the last part: 

If `G(x) = (f(x))^9`

Then using the chain rule: 

`G'(x) = 9[f(x)}^8 * f'(x)`

Now we can apply ourselves by substituting 'a' for 'x':

`G'(a) = 9 [f(a)]^8 * f'(a)`  

Now substitute eq. 2.2 and eq. 2.3 in the above equation: 

`G'(a) = 9* (2)^8 * 12 = 27648`

SUMMARY: 

F'(a) = 216

G'(a) = 27648