What is the first derivative of ln|x| ?

The first derivative of ln|x| is 1/x

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derivative of ln|x|

derivative of

ln ∣ x ∣

\frac{1}{x}

Solution steps

\frac{d}{d x} (ln ∣ x ∣)

Apply the chain rule:

\frac{1}{∣ x ∣} \frac{d}{d x} (∣ x ∣)

= \frac{1}{∣ x ∣} \frac{d}{d x} (∣ x ∣)

\frac{d}{d x} (∣ x ∣) = \frac{x}{∣ x ∣}

= \frac{1}{∣ x ∣} \cdot \frac{x}{∣ x ∣}

Simplify

\frac{1}{∣ x ∣} \cdot \frac{x}{∣ x ∣} : \frac{1}{x}

= \frac{1}{x}