A) \[\frac{3}{5}\log \,2\]
B) \[\frac{2}{5}\log \,2\]
C) \[-\frac{3}{2}\log \,2\]
D) \[\log \,2\left( \frac{-1}{10} \right)\]
Correct Answer: D
Solution :
[d] Given expression can be written as \[y={{\tan }^{-1}}\left[ \frac{{{2}^{x}}(2-1)}{1+{{2}^{x}}{{.2}^{x+1}}} \right]={{\tan }^{-1}}\left[ \frac{{{2}^{x+1}}-{{2}^{x}}}{1+{{2}^{x}}{{.2}^{x+1}}} \right]\] \[={{\tan }^{-1}}({{2}^{x+1}})-ta{{n}^{-1}}({{2}^{x}})\] \[\Rightarrow \frac{dy}{dx}=\frac{{{2}^{x+1}}\log \,2}{1+{{2}^{2(x+1)}}}-\frac{{{2}^{x}}\log 2}{1+{{2}^{2x}}}\] \[\therefore {{\left( \frac{dy}{dx} \right)}_{x=0}}=(log\,\,2)\left( \frac{2}{5}-\frac{1}{2} \right)=\log 2\left( -\frac{1}{10} \right)\]
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