Question

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

Derivatives
Find the derivatives of the functions. $$\displaystyle{f{{\left({x}\right)}}}={\ln}{\left|{\ln{{x}}}\right|}$$

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
$$\displaystyle{f}'{\left({x}\right)}={\frac{{{1}}}{{{\ln{{x}}}}}}{\frac{{{1}}}{{{x}}}}={\frac{{{1}}}{{{x}{\ln{{x}}}}}}$$