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: 

 ... Eq. 1.1

 ... Eq. 1.2

Additionally we are given :

  ... Eq. 2.1

 ... Eq. 2.2 

 ... Eq. 2.3

 ... Eq. 2.4

We are required to find: 

And 

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

If 

 Then using the chain rule :

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

We can now substitute Eq. 2.4 and Eq. 2.1

 

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

If 

Then using the chain rule: 

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

 

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

SUMMARY: 

F'(a) = 216

G'(a) = 27648