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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Differentiation

NDA 2018 · Mathematics · Differentiation

If y= ^ -1 ( 5-2 x 2+5 x ), then what is (d y)/(d x) equal to?

If y=tan⁡−1(5−2tan⁡x2+5tan⁡x),y=\tan ^{-1} \left(\frac{5-2 \tan \sqrt{x}}{2+5 \tan \sqrt{x}}\right),then what is dydx\frac{d y}{d x}equal to?

Options

  1. A−12x-\frac{1}{2 \sqrt{\mathrm{x}}}
  2. B1
  3. C-1
  4. D12x\frac{1}{2 \sqrt{\mathrm{x}}}

Correct answer

A. −12x-\frac{1}{2 \sqrt{\mathrm{x}}}

Step-by-step solution

y=tan⁡−1(5−2tan⁡x2+5tan⁡x)=tan⁡−1(52−tan⁡x1+(52)tan⁡x)=tan⁡−152−tan⁡−1tan⁡x=tan⁡−152−xdydx=−12x\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}
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