A) \[{{(x-2)}^{x}}\log (x-2)+x{{(x-2)}^{x-1}}\]
B) \[{{(x-2)}^{x}}\log (x-2)+{{(x-2)}^{x}}\]
C) \[{{(x-2)}^{x-1}}\log (x-1)+x{{(x-2)}^{x-1}}\]
D) None of the above
Correct Answer: A
Solution :
\[y={{(x-2)}^{x}}\] Taking log on both sides \[log\,y=x\,log(x-2)\] On differentiating w.r.t.\[x,\] \[\frac{1}{y}\frac{dy}{dx}=x.\frac{1}{x-2}+\log (x-2)\] \[\Rightarrow \] \[\frac{dy}{dx}=y\left[ \frac{x}{x-2}+\log (x-2) \right]\] \[\Rightarrow \] \[\frac{dy}{dx}={{(x-2)}^{x}}[x{{(x-2)}^{-1}}+\log (x-2)]\] \[\Rightarrow \] \[\frac{dy}{dx}=x{{(x-2)}^{x-1}}+{{(x-2)}^{x}}+\log (x-2)\]
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