Question

Find the derivatives of the functions. f(x)=\ln |\ln x|

Derivatives
ANSWERED
asked 2021-06-21
Find the derivatives of the functions. \(\displaystyle{f{{\left({x}\right)}}}={\ln}{\left|{\ln{{x}}}\right|}\)

Answers (1)

2021-06-22
The function we want to differentiate is
\(\displaystyle{f{{\left({x}\right)}}}={\ln}{\left|{\ln{{x}}}\right|}\)
Recall the generalized rule for the derivative of a natural logarithm of an absolute value:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\ln}{\left|{u}\right|}\right]}={\frac{{{1}}}{{{u}}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\)
Now we get
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{1}}}{{{\ln{{x}}}}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\ln{{x}}}\right)}\)
Recall that
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\ln{{x}}}\right)}={\frac{{{1}}}{{{x}}}}\)
Thus
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{1}}}{{{\ln{{x}}}}}}{\frac{{{1}}}{{{x}}}}={\frac{{{1}}}{{{x}{\ln{{x}}}}}}\)
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