Expert: Paul Klarreich - 5/7/2006

Question
Q:  Can you take the natural log of a natural log?

1. Kelly
2. Calculus
Problem: f(x) = ln[ln(x)]

3. My Answer f'x = [ln(1/x)+ln(lnx)]

Thank you

Answer
Hi, Kelly,
You wrote:  
Subject:  natural logs
Question:  Q: Can you take the natural log of a natural log?

>> Yes, but as long as neither one uses pesticides.

1. Kelly
2. Calculus
Problem: f(x) = ln[ln(x)]

3. My Answer f'x = [ln(1/x)+ln(lnx)]

OK, OK, back to serious math now.

Sorry -- you made a mistake.

ln[ln(x)] is NOT a product of two functions; it is a composition.  For compositions of functions you use the Chain Rule.  It really should be called the Composition Rule, because that's what it deals with.

Write:

y = ln[u],  where  u = ln(x)

dy    1
-- = ---
du    u

du    1
-- = ---
dx    x

dy   dy du    1   1
-- = -- -- = --- ---
dx   du dx    u   x
   1    1       1
= ----- --- = -------
 ln(x)  x    x ln(x)