AP EAMCET · Maths · Differentiation
For \(x \lt 0, \frac{d}{d x}\left[|x|^x\right]=\)
- A \((-x)^x[-1+\log (-x)]\)
- B \((-x)^{\mathrm{x}}[1+\log (-x)]\)
- C \((-x)^{\mathrm{x}}[1-\log (-x)]\)
- D \((-x)^{\mathrm{x}}[-1-\log (-x)]\)
Answer & Solution
Correct Answer
(B) \((-x)^{\mathrm{x}}[1+\log (-x)]\)
Step-by-step Solution
Detailed explanation
\(|x|=-x \quad \forall x \lt 0 . \Rightarrow|x|^x=(-x)^x \Rightarrow y=(-x)^x\) \(\log y=x \log (-x) \Rightarrow \frac{y^{\prime}}{y}=1+\log (-x)\) \(\frac{d y}{d x}=(-x)^x(\log (-x)+1)\)
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