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