The derivative of \( |x| \) with respect to \( x \), where \( x\in R\sim\{0\} \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{x}{|x|} \)
For x > 0, |x| = x, so the derivative is 1.
For x < 0, |x| = −x, so the derivative is −1.
Both cases are given by one expression:
\( \displaystyle \frac{d}{dx}|x|=\frac{x}{|x|}\quad(x\ne 0) \)
Why the other options are incorrect
Option (1) 1 — Incorrect. This is the derivative only for x > 0.
Option (3) \( -1 \) — Incorrect. This is the derivative only for x < 0.
Option (4) \( \displaystyle\frac{1}{x} \) — Incorrect. This is the derivative of \( \log|x| \), not of |x|.