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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Differentiation

NDA 2023 · Mathematics · Differentiation

Paragraph: Let f(x)=| x|, x ≠ 1 Question: What is the derivative of...

Paragraph: Let f(x)=∣ln⁡x∣,x≠1f(x)=|\ln x|, x \neq 1
Question: What is the derivative of f∘f(x)f {}^{\circ} f(x), where 1<x<21 < \mathrm{x} < 2 ?

Options

  1. A1ln⁡x\frac{1}{\ln x}
  2. B1xln⁡x\frac{1}{x \ln x}
  3. C−1ln⁡x-\frac{1}{\ln x}
  4. D−1xln⁡x-\frac{1}{x \ln x}

Correct answer

D. −1xln⁡x-\frac{1}{x \ln x}

Step-by-step solution

Since, 1<x<2⇒∣ln⁡x∣=ln⁡x1 < x < 2 \Rightarrow|\ln x|=\ln x and 0<ln⁡x<ln⁡2⇒∣ln⁡x∣=−ln⁡x0 < \ln x < \ln 2 \Rightarrow|\ln x|=-\ln x Hence y=\mathrm{y}= fof (x)=∣ln⁡∣∣ln⁡x∣=−ln⁡(ln⁡x)(x)=|\ln ||\ln x|=-\ln (\ln x), for ∣<x<2y′=−1ln⁡x×1x⇒y′=−1xln⁡x\mid < x < 2y^{\prime}=\frac{-1}{\ln x} \times \frac{1}{x} \Rightarrow y^{\prime}=\frac{-1}{x \ln x}
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