Q227 / 270 · NDA 2018 (Phase 2) · Mathematics

If $y=\tan ^{-1}\left(\frac{5-2 \tan \sqrt{x}}{2+5 \tan \sqrt{x}}\right)$, then what is $\frac{d y}{d x}$ equal to?

  • Correct answer
    A
    $-\frac{1}{2 \sqrt{\mathrm{x}}}$
  • B
    1
  • C
    -1
  • D
    $\frac{1}{2 \sqrt{\mathrm{x}}}$

Solution

$\begin{aligned} & y=\tan ^{-1}\left(\frac{5-2 \tan \sqrt{x}}{2+5 \tan \sqrt{x}}\right) \\ & =\tan ^{-1}\left(\frac{\frac{5}{2}-\tan \sqrt{x}}{1+\left(\frac{5}{2}\right) \tan \sqrt{x}}\right) \\ & =\tan ^{-1} \frac{5}{2}-\tan ^{-1} \tan \sqrt{x} \\ & =\tan ^{-1} \frac{5}{2}-\sqrt{x} \\ & \frac{d y}{d x}=-\frac{1}{2 \sqrt{x}}\end{aligned}$

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

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