d dx ^ - 1 x - x^ - 1 x + x^ - 1 = - Vidyadip

MCQ

${d \over {dx}}{\cos ^{ - 1}}{{x - {x^{ - 1}}} \over {x + {x^{ - 1}}}} =$

A
${1 \over {1 + {x^2}}}$
B
${{ - 1} \over {1 + {x^2}}}$
C
${2 \over {1 + {x^2}}}$
✓
${{ - 2} \over {1 + {x^2}}}$

✓

Answer

Correct option: D.

${{ - 2} \over {1 + {x^2}}}$

d
(d) Putting $x = \cot \theta $

$y = {\cos ^{ - 1}}\left( {\frac{{x - {x^{ - 1}}}}{{x + {x^{ - 1}}}}} \right) = {\cos ^{ - 1}}\left( {\frac{{{x^2} - 1}}{{{x^2} + 1}}} \right)$

$ = {\cos ^{ - 1}}(\cos 2\theta ) = 2\theta $

$\Rightarrow \frac{{dy}}{{dx}} = \frac{{ - 2}}{{1 + {x^2}}}$.

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