A) ? 1
B) 1
C) \[\log \,\,2\]
D) \[-\log \,2\]
Correct Answer: A
Solution :
\[f(x)={{\cot }^{-1}}\left( \frac{{{x}^{x}}-{{x}^{-x}}}{2} \right)\]; Put \[{{x}^{x}}=\tan \theta \] \ \[y=f(x)={{\cot }^{-1}}\left( \frac{{{\tan }^{2}}\theta -1}{2\tan \theta } \right)\] = \[{{\cot }^{-1}}(-\cot 2\theta )\] = \[\pi -{{\cot }^{-1}}(\cot 2\theta )\] Þ y = \[\pi -2\theta \] = \[\pi -2{{\tan }^{-1}}({{x}^{x}})\] \[\frac{dy}{dx}=\frac{-2}{1+{{x}^{2x}}}.{{x}^{x}}(1+\log x)\] Þ \[{f}'(1)=-1\].
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