Differentiation Question 456
Question: $ \frac{d}{dx}({x^{{\log _{e}}x}})= $
[MP PET 1993]
Options:
A) $ 2{x^{({\log _{e}}x-1)}}.{\log _{e}}x $
B) $ {x^{({\log _{e}}x-1)}} $
C) $ \frac{2}{x}{\log _{e}}x $
D) $ {x^{({\log _{e}}x-1)}}.{\log _{e}}x $
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Answer:
Correct Answer: A
Solution:
Let $ y={x^{{\log _{e}}x}} $
Therefore $ {\log _{e}}y={\log _{e}}x{\log _{e}}x={{({\log _{e}}x)}^{2}} $
Therefore $ \frac{1}{y}\frac{dy}{dx}=2{\log _{e}}x.\frac{1}{x} $
$ \therefore \frac{dy}{dx}=2{x^{({\log _{e}}x-1)}}{\log _{e}}x $ .
BETA
