Q220 / 270 · NDA 2022 (Phase 1) · Mathematics

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

  • A
    $\frac{1}{2}$
  • Correct answer
    B
    1
  • C
    2
  • D
    4

Solution

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

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

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