A) \[\frac{-1}{2\sqrt{1-{{x}^{4}}}}\]
B) \[\frac{1}{2\sqrt{1-{{x}^{4}}}}\]
C) \[\frac{-x}{\sqrt{1-{{x}^{4}}}}\]
D) \[\frac{x}{\sqrt{1-{{x}^{4}}}}\]
Correct Answer: C
Solution :
Putting \[{{x}^{2}}=\cos 2\theta \], we have \[\frac{d}{dx}\left[ {{\cos }^{-1}}\sqrt{\frac{1+{{x}^{2}}}{2}} \right]=\frac{d}{dx}[{{\cos }^{-1}}\cos \theta ]\] \[=\frac{d}{dx}[\theta ]=\frac{d}{dx}\left[ \frac{1}{2}{{\cos }^{-1}}{{x}^{2}} \right]=\frac{-x}{\sqrt{1-{{x}^{4}}}}\].
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