Suppose g(x)=ln(ln(ln(f(x)))), f(4)=A, and f'(4)=B. Find the derivative dg/dx||||x=4. g'(4)=?

Question

I need help pleaseee

Answer 1

To differentiate, you have to use the chain rule several times:

 

D_x g(x) = [ 1 / ln (ln (f(x))) ] * D_x ln (ln (f(x))) = 

[ 1 / ln (ln (f(x))) ] * [ 1 / ln (f(x)) ] * D_x ln (f(x)) = 

[ 1 / ln (ln (f(x))) ] * [ 1 / ln (f(x)) ] * [ 1 / f(x) ] [ D_x f(x) ] =

[ 1 / ln (ln (f(x))) ] * [ 1 / ln (f(x)) ] * [ 1 / f(x) ] [ f'(x) ]

 

Now plug in the values for f(4) and f'(4)...

 

[ 1 / ln (ln (f(x))) ] * [ 1 / ln (f(x)) ] * [ 1 / f(x) ] [ f'(x) ] =

[ 1 / ln (ln (A)) ] * [ 1 / ln (A) ] * [ 1 / A ] [ B ]

 

That's about it, but you can make it look at little simpler...

 

                            B

g'(4) =     --------------------------

                ln (ln (A)) * ln (A) * A