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GATE 2021 BT – Question 18

Engineering Mathematics · Calculus: Limits, continuity, differentiability, partial derivatives, maxima and minima · 1 mark · Multiple choice

$d[\ln(2x)]/dx$ is

  1. $1/(2x)$
  2. $1/x$
  3. $1/2$
  4. $x$

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Correct answer: (B) $1/x$

Explanation

**Chain rule.** If $u=2x$, then $\dfrac{d}{dx}\ln u=\dfrac1u\cdot\dfrac{du}{dx}$:
$$\frac{d}{dx}\ln(2x)=\frac1{2x}\cdot2=\frac1x .$$

**Alternative check.** $\ln(2x)=\ln2+\ln x$, and the derivative of the constant $\ln2$ is 0, so the derivative is $\dfrac1x$.

Answer **$1/x$** (B).