Explanation

(c)
We have, $y = \log{|x}$|$= \left\{ \begin{matrix}
\log{x,}\ \ \ \ \ \ \ \ \ \ \ \ \ \ x > 0 \\
\log{( - x),\ \ \ \ \ \ \ \ x < 0}
\end{matrix} \right.\ \ \ \ \ \ \$
$\therefore\ \frac{dy}{dx} = \left\{ \begin{matrix}
\frac{1}{x},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x > 0 \\
\frac{1}{- x}( - 1) = \frac{1}{x},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x < 0\ \ \ \ \ \ \ \ \ \
\end{matrix} \right.\ \ \ \ \ \ \ \$
$\Rightarrow \ \frac{dy}{dx}$ = $\frac{1}{x},x \neq 0$