What is the derivative of #Ln(ln(ln(2x)))#?

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Answer 1 · 2016-09-05T02:00:06

#1/(( ln (ln (2x)))(ln (2x)(x))#, #x>e/2#

For real logarithms:

ln (2x) is real, for x> 0#

ln ( ln (2x)) is real, for ln (2x) > 0, So, ln (2x) > 0. And so, x > 1/2.

ln (ln (ln (2x))) is real, for ln(ln(2x))> 0. So,

ln (ln (2x)) >0 #to# ln (2x) > 1 # to # 2x > e # to # x > e/2.

All are real, for x > e/2.

In succession, apply function rule.

#(ln(ln(ln(2x))))'#

#=1/(ln(ln(2x)))(ln(ln(2x))'#

#=1/((ln(ln(2x)))(ln(ln(2x))# #(ln(2x))'#

#=1/((ln(ln(2x)))(ln(ln(2x)))# #1/(2x)(2x)'#

#=1/((ln(ln(2x)))(ln(2x)(x))#, #x>e/2#
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