A) \[\frac{{{y}^{2}}-xy\,\log \,y}{{{x}^{2}}-xy\,\log \,x}\]
B) \[\frac{{{y}^{2}}+xy\,\log \,y}{{{x}^{2}}+xy\,\log \,x}\]
C) \[\frac{{{y}^{2}}-xy\,\log \,x}{{{x}^{2}}-xy\,\log \,y}\]
D) \[\frac{{{y}^{2}}+xy\,\log \,y}{{{x}^{2}}-xy\,\log \,x}\]
Correct Answer: A
Solution :
Given, \[{{x}^{y}}={{y}^{x}}\] Taking log both sides, we get \[y\,\log \,x=x\,\log \,y\] On differentiating w.r.t.x, we get \[\frac{y}{x}+\log x\frac{dy}{dx}=\frac{x}{y}\frac{dy}{dx}+\log y\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{\log y-\frac{y}{x}}{\log x-\frac{x}{y}}\] \[=\frac{xy\,\log \,y-{{y}^{2}}}{xy\,\log \,x-{{x}^{2}}}\] \[=\frac{{{y}^{2}}-xy\,\log \,y}{{{x}^{2}}-xy\,\log \,x}\]
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