A) \[{{\left( 1+\frac{1}{x} \right)}^{x}}\left[ \log \left( 1+\frac{1}{x} \right)-\frac{1}{1+x} \right]\]
B) \[{{\left( 1+\frac{1}{x} \right)}^{x}}\left[ \log \left( 1+\frac{1}{x} \right) \right]\]
C) \[{{\left( x+\frac{1}{x} \right)}^{x}}\left[ \log (x-1)-\frac{x}{x+1} \right]\]
D) \[{{\left( 1+\frac{1}{x} \right)}^{x}}\left[ \log \left( 1+\frac{1}{x} \right)+\frac{1}{1+x} \right]\]
Correct Answer: A
Solution :
\[y={{\left( 1+\frac{1}{x} \right)}^{x}}\Rightarrow \log y=x\log \left( 1+\frac{1}{x} \right)\] \[\Rightarrow \frac{1}{y}\frac{dy}{dx}=\log \left( 1+\frac{1}{x} \right)-\frac{1}{1+x}\] Þ \[\frac{dy}{dx}={{\left( 1+\frac{1}{x} \right)}^{x}}\left[ \log \left( 1+\frac{1}{x} \right)-\frac{1}{1+x} \right]\].
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