A) \[\frac{3}{5}\log \,\,2\]
B) \[\frac{2}{5}\log \,\,2\]
C) \[-\frac{3}{2}\log \,2\]
D) None of these
Correct Answer: D
Solution :
Given expression can be written as \[y={{\tan }^{-1}}\left[ \frac{{{2}^{x}}(2-1)}{1+{{2}^{x}}{{.2}^{x+1}}} \right]={{\tan }^{-1}}\,\,\left[ \frac{{{2}^{x+1}}-{{2}^{x}}}{1+{{2}^{x}}{{.2}^{x+1}}} \right]\] \[=\,\,{{\tan }^{-1}}({{2}^{x+1}})-{{\tan }^{-1}}({{2}^{x}})\] \[\Rightarrow \,\,\,\frac{dy}{dx}=\frac{{{2}^{x+1}}\log \,2}{1+{{2}^{2(x+1)}}}-\frac{{{2}^{x}}\log 2}{1+{{2}^{2x}}}\] \[\therefore \,\,{{\left( \frac{dy}{dx} \right)}_{x=0}}=(log\,2)\left( \frac{2}{5}-\frac{1}{2} \right)=\log \,\,2\left( -\,\frac{1}{10} \right)\]
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