A) \[{{(1+x)}^{x}}\left[ \frac{x}{1+x}+\log ex \right]\]
B) \[\frac{x}{1+x}+\log (1+x)\]
C) \[{{(1+x)}^{x}}\left[ \frac{x}{1+x}+\log (1+x) \right]\]
D) None of these
Correct Answer: C
Solution :
\[y={{(1+x)}^{x}}\] Taking log on both sides, \[\log y=x\log (1+x)\] Differentiating w.r.t. x, we get \[\frac{1}{y}\frac{dy}{dx}=\log (1+x)+x\frac{1}{(1+x)}\] Thus \[\frac{dy}{dx}={{(1+x)}^{x}}\left[ \frac{x}{1+x}+\log (1+x) \right]\]
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