Q219 / 270 · NDA 2022 (Phase 1) · Mathematics

If $x^y y^x=1$, then what is $\frac{d y}{d x}$ at $(1,1)$ equal to?

  • Correct answer
    A
    -1
  • B
    0
  • C
    1
  • D
    4

Solution

$\mathrm{x}^{\mathrm{y}} \mathrm{y}^{\mathrm{x}}=1$ Taking log both sides $\begin{aligned} & y \log x+x \log y=0 \\ & y^{\prime} \log x+\frac{y}{x}+\log y+\frac{x}{y} y^{\prime}=0\end{aligned}$ put $\mathrm{x}=1, \mathrm{y}=1$ $\begin{aligned} & \mathrm{y}^{\prime} \log (1)+1+\log (1)+\mathrm{y}^{\prime}=0 \\ & 0+1+0+\mathrm{y}^{\prime}=0 \Rightarrow \mathrm{y}^1=-1\end{aligned}$

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

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