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