Question

# Find the derivatives of the functions. f(x)=\frac{1}{\ln |x|}

Derivatives
Find the derivatives of the functions. $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{\ln}{\left|{x}\right|}}}}$$

2021-06-28
First, let's rewrite the function a bit.
$$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{\ln}{\left|{x}\right|}}}}={\left({\ln}{\left|{x}\right|}\right)}^{{-{1}}}$$
Using the chain rule, we get
$$\displaystyle{f}'{\left({x}\right)}={\left[{\left({\ln}{\left|{x}\right|}\right)}^{{-{1}}}\right]}'=-{\left({\ln}{\left|{x}\right|}\right)}^{{-{2}}}{\left({\ln}{\left|{x}\right|}\right)}'$$
Recall that
$$\displaystyle{\left({\ln}{\left|{x}\right|}\right)}'={\frac{{{1}}}{{{x}}}}$$
Hence
$$\displaystyle{f}'{\left({x}\right)}=-{\left({\ln}{\left|{x}\right|}\right)}^{{-{2}}}{\frac{{{1}}}{{{2}}}}=-{\frac{{{1}}}{{{x}{\left({\ln}{\left|{x}\right|}\right\rbrace}^{{{2}}}}}}$$