(A) We know that $\log |x|$ is defined as:$\log |x| = \begin{cases} \log x, & \text{if } x > 0 \\ \log(-x), & \text{if } x < 0 \end{cases}$Case $1$: I

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(A) We know that $\log |x|$ is defined as:$\log |x| = \begin{cases} \log x, & \text{if } x > 0 \\ \log(-x), & \text{if } x Case $1$: If $x > 0$,then $\frac{d}{dx}(\log x) = \frac{1}{x}$.Case $2$: If $x Since the derivative is $\frac{1}{x}$ in both cases,we conclude that $\frac{d}{dx} \log |x| = \frac{1}{x}$ for all $x \ne 0$.