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Question: If \(f(x) = |\log x|,\) then for \(x \neq 1,f^{'}(x)\) equals...

If f(x)=∣log⁡x∣,f(x) = |\log x|, then for x≠1,f′(x)x \neq 1,f^{'}(x) equals

A

1x\frac{1}{x}

B

1∣x∣\frac{1}{|x|}

C

−1x\frac{- 1}{x}

D

None of these

Answer

None of these

Explanation

Solution

$f(x) = |\log x| = \left{ \begin{matrix}

  • \log x, & \text{if }0 < x < 1 \ \log x, & \text{if }x \geq 1 \end{matrix} \right.\ $

⇒ $f^{'}(x) = \left{ \begin{matrix}

  • \frac{1}{x}, & \text{if }0 < x < 1 \ \frac{1}{x}, & \text{if }x > 1 \end{matrix} \right.\ $.

Clearly f′(1−)=−1f^{'}(1^{-}) = - 1 and f′(1+)=1f^{'}(1^{+}) = 1,

∴\therefore f′(x)f^{'}(x) does not exist at x=1x = 1