Q103 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

For $x \lt 0, \frac{d}{d x}\left[|x|^x\right]=$

  • A
    $(-x)^x[-1+\log (-x)]$
  • Correct answer
    B
    $(-x)^{\mathrm{x}}[1+\log (-x)]$
  • C
    $(-x)^{\mathrm{x}}[1-\log (-x)]$
  • D
    $(-x)^{\mathrm{x}}[-1-\log (-x)]$

Solution

$|x|=-x \quad \forall x \lt 0 . \Rightarrow|x|^x=(-x)^x \Rightarrow y=(-x)^x$ $\log y=x \log (-x) \Rightarrow \frac{y^{\prime}}{y}=1+\log (-x)$ $\frac{d y}{d x}=(-x)^x(\log (-x)+1)$

Hence, the correct answer is option: 'B'.

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