Answer to Question #96194 in Calculus for Emily

Question

Suppose g(x)=ln(ln(ln(f(x)))), f(9)=A and f’(9)=B Find the derivative dg/dx, f=9
What is g’(9)?

Answer 1

We need to find the Derivative of g(x) at x= 9.

We know the derivative of ln (x) is 1/x.

So, derivative of ln (f(x) ) is [ 1/f(x)] * derivative of f(x)

Given function g(x) = ln ( ln ( ln (f(x))))

Differentiate with respect to x

g' (x) = ( 1 / ln (ln (f(x))) * (1/ ln (f(x)) * (1 / f(x) ) * f ' (x)

Plug x = 9

g' (9) = ( 1 / ln (ln (f(9))) * (1/ ln (f(9)) * (1 / f(9) ) * f ' (9)

g' (9) = ( 1 / ln (ln (A)) * (1/ ln (A) * (1 / A ) * B

Since f(9) = A and f ' (9 ) = B.

Answer: g' (9) = B / ( A* ln (A) * ln (ln A) ).