Given the function $f$, evaluate $f'$

Question

Let $$f(x) = \ln \left(\frac{1}{|x|}\right).$$ Evaluate $f'$.

The answer key is $f' = \frac{-1}{x\ln10}$. I have no idea how to start because of $|x|$. Any help is appreciated.

Answer 1

Hint: You can break it into cases: $f(x)=\begin{cases} \ln(\frac1x), x\gt0\\\ln(-\frac1x), x\lt0\end{cases}$.

Use the chain rule.

Answer 2

Hint. Let $g(x)=|x|$, then $g'(x)=\mathop{\rm sgn} x$ for $x\neq 0$; but then you will need to use some properties for $|x|$ and $\mathop{\rm sgn} x$ at the very end, such as $|x|=x \cdot \mathop{\rm sgn} x$ and $\dfrac{\mathop{\rm sgn} x}{x}= \dfrac{1}{x \cdot \mathop{\rm sgn} x}$.