Q102 / 160 · AP EAMCET 2024 (23 May Shift 1) · Mathematics

If $y=\sinh ^{-1}\left(\frac{1-x}{1+x}\right)$ then $\frac{d y}{d x}$

  • Correct answer
    A
    $\frac{-\sqrt{2}}{|1+x| \sqrt{1+x^2}}$
  • B
    $\frac{-1}{(1+x) \sqrt{x}}$
  • C
    $\frac{1}{\left(1+x^2\right) \sqrt{1+x}}$
  • D
    $\frac{-\sqrt{2}}{(1+x) \sqrt{1-x}}$

Solution

$y=\sin h^{-1}\left(\frac{1-x}{1+x}\right)$ $\begin{aligned} & \therefore \frac{d y}{d x}=\frac{1}{\sqrt{\left(\frac{1-x}{1+x}\right)^2+1}} \times \frac{-(1+x)-(1-x)}{(1+x)^2} \\ & \Rightarrow \frac{d y}{d x}=\frac{-2}{|1+x| \sqrt{2+2 x^2}}=\frac{-\sqrt{2}}{|1+x| \sqrt{1+x^2}}\end{aligned}$

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

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