Answer
$\tan x$
Work Step by Step
Let us suppose that $f(x)= \ln [|\sec x|]$
Use reciprocal identity $\sec x=\dfrac{1}{\cos x}$
We differentiate both sides with respect to $x$.
$f^{\prime}(x)=\dfrac{d}{dx} [\ln \dfrac{1}{\cos x}] \\=-(\dfrac{1}{\cos x} \dfrac{d}{dx} [\cos x]$
Simplify to obtain:
$f^{\prime}(x)=-(\dfrac{1}{\cos x} (-\sin x)=\tan x$
