Answer
$ \displaystyle \frac{2\ln|x|}{x}$
Work Step by Step
$f(x)$ is a composite function,
f(x)=u(v(x)).
The last calculation in this composition is squaring so
$u(x)=x^{2}, \displaystyle \frac{du}{dx}=2x$
$v=\ln|x|$.
For $\displaystyle \frac{dv}{dx}$, apply Derivatives of logarithms of absolute values:
$\displaystyle \frac{d}{dx} [\displaystyle \ln|x|]=\frac{1}{x}$,
Apply the chain rule for f(x)=u(v(x))
$\displaystyle \frac{d}{dx}[u(v(x))]=\frac{du}{dv}\frac{dv}{dx}$
$=2v\displaystyle \cdot\frac{1}{x}$
$=2\displaystyle \ln|x|\cdot\frac{1}{x}$
$=\displaystyle \frac{2\ln|x|}{x}$
