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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Differentiation

NDA 2022 · Mathematics · Differentiation

If y=(x^x)^x , then what is the value of (d y)/(d x) at x=1 ?

If y=(xx)xy=\left(x^x\right)^x, then what is the value of dydx\frac{d y}{d x} at x=1x=1 ?

Options

  1. A12\frac{1}{2}
  2. B1
  3. C2
  4. D4

Correct answer

B. 1

Step-by-step solution

y=(xx)x=(x)x2\left(x^x\right)^x=(x)^{x^2} Taking log both side log⁡y=x2log⁡x\log y=x^2 \log x1ydydx=2x⋅log⁡x+x2⋅1x1ydydx=2xlog⁡x+x\begin{aligned} & \frac{1}{y} \frac{d y}{d x}=2 x \cdot \log x+x^2 \cdot \frac{1}{x} \\ & \frac{1}{y} \frac{d y}{d x}=2 x \log x+x\end{aligned}$\begin{aligned} & \text{ put } x=1, y=1 \\ & \frac{d y}{d x}=2 \log (1)+1=1\end{aligned}
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