Solve for g(8) and g'(8) in f(g(x)) = x using the Chain Rule

[fiziksfun]

ok so f(g(x)) = x, for all x.

f(3)=8
f'(3)=9

what are the values of g(8) and g'(8)

ok, so g(8) = 3

because f(g(8)) must equal 8, and f(3) = 8, so g(x) must equal three.

however, i have NO idea how to do g'(x)

i was thinking of using the chain rule, but this gets me nowhere..help!

f'(g(x))*g'(x) = 8 ?? is this correct?? then wouldn't g'(x) = 1 ??

 

[Vid]

chain rule

 

[HallsofIvy]

ok so f(g(x)) = x, for all x.

f(3)=8
f'(3)=9

what are the values of g(8) and g'(8)

ok, so g(8) = 3

because f(g(8)) must equal 8, and f(3) = 8, so g(x) must equal three.

however, i have NO idea how to do g'(x)

i was thinking of using the product rule, but this gets me nowhere..help!

f'(g(x))*g'(x) = 8 ?? is this correct?? then wouldn't g'(x) = 1 ??

chain rule

What fiziksfun wrote in his last line is the chain rule, not the product rule.
If (f(g(x))= x then f'(g(x))*g'(x)= (x)'= 1, not 8.

 

[fiziksfun]

the chain rule doesn't get me anywhere :[

 

[fiziksfun]

how do you know (x)' is equal to 1 ?

 

[Vid]

Yea, I saw product rule in his post and just skipped over the symbols.

 

[fiziksfun]

oh wait, is it because d/dx(x) = 1 ?? YAY!

 

[HallsofIvy]

Wow, that was fast!

I must admit that when you asked how I knew that the derivative of x was 1, I started to reach for my 2 by 4!