A) \[\frac{1}{1+{{\tan }^{-1}}x}\]
B) \[\frac{-1}{1+{{\tan }^{-1}}x}\]
C) \[\frac{1}{{{(1+{{\tan }^{-1}}x)}^{^{2}}}}\]
D) \[\frac{-1}{2\,{{(1+{{\tan }^{-1}}x)}^{2}}}\]
Correct Answer: C
Solution :
The differential coefficient of \[\frac{{{\tan }^{-1}}x}{1+{{\tan }^{-1}}x}\] with respect to \[{{\tan }^{-1}}x=\frac{\frac{d}{dx}\left( \frac{{{\tan }^{-1}}x}{1+{{\tan }^{-1}}x} \right)}{\frac{d}{dx}({{\tan }^{-1}}x)}\] \[=\frac{(1+{{\tan }^{-1}}x)\frac{d}{dx}{{\tan }^{-1}}x-{{\tan }^{-1}}x\frac{d}{dx}(1+{{\tan }^{-1}}x)}{{{(1+{{\tan }^{-1}}x)}^{2}}.\frac{d}{dx}{{\tan }^{-1}}x}\] \[=\frac{1}{{{(1+{{\tan }^{-1}}x)}^{2}}}\].
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