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NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Differentiation

NDA 2014 · Mathematics · Differentiation

What is the derivative of (1+ x)/(1- x) ?

What is the derivative of \sqrt{\frac{1+\cos x}{1-\cos x}}$ ?

Options

  1. A12sec⁡2x2\frac{1}{2} \sec ^2 \frac{x}{2}
  2. B−12cosec⁡2x2-\frac{1}{2} \operatorname{cosec}^2 \frac{x}{2}
  3. C−cosec⁡2x2-\operatorname{cosec}^2 \frac{x}{2}
  4. DNone of the above

Correct answer

B. −12cosec⁡2x2-\frac{1}{2} \operatorname{cosec}^2 \frac{x}{2}

Step-by-step solution

y=1+cos⁡x1−cos⁡x=2cos⁡2x22sin⁡2x2=cos⁡2x2sin⁡2x2=cos⁡x2sin⁡x2=cot⁡x2\begin{aligned} y & =\sqrt{\frac{1+\cos x}{1-\cos x}}=\sqrt{\frac{2 \cos ^2 \frac{x}{2}}{2 \sin ^2 \frac{x}{2}}} \\ & =\sqrt{\frac{\cos ^2 \frac{x}{2}}{\sin ^2 \frac{x}{2}}}=\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}=\cot \frac{x}{2} \end{aligned} Differentiating both sides with respect to xx. dydx=−12cosec⁡2x2.\frac{d y}{d x}=-\frac{1}{2} \operatorname{cosec}^2 \frac{x}{2} .
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