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Continuity And Differentiability

Question
CBSEENMA12034860

Differentiate the following functions w.r.t. x :
left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log space straight x end exponent

Solution
Let space straight y equals left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log space straight x end exponent
Put space left parenthesis log space straight x right parenthesis to the power of straight x equals straight u comma space straight x to the power of logx equals straight v
therefore space straight y equals straight u plus straight v
therefore space dy over dx equals du over dx plus dv over dx space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
Now space straight u equals left parenthesis log space straight x right parenthesis to the power of straight x
therefore space log space straight u equals log left parenthesis log space straight x right parenthesis to the power of straight x
therefore space log space straight u equals straight x. log left parenthesis log space straight x right parenthesis
therefore space 1 over straight u du over dx equals straight x. open parentheses fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close parentheses plus log left parenthesis log space straight x right parenthesis.1
therefore space du over dx equals straight u open square brackets fraction numerator 1 over denominator log space straight x end fraction plus log left parenthesis log space straight x right parenthesis close square brackets
therefore space du over dx equals left parenthesis log space straight x right parenthesis to the power of straight x open square brackets fraction numerator 1 over denominator log space straight x end fraction plus log left parenthesis log space straight x right parenthesis close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
Also space straight v equals straight x to the power of log space straight x end exponent
therefore space log space straight v equals log left parenthesis straight x to the power of log space straight x end exponent right parenthesis
space space space space space space space space space space space space space equals log space straight x. log space straight x equals left parenthesis log space straight x right parenthesis squared
therefore space 1 over straight v dv over dx equals left parenthesis 2 log space straight x right parenthesis. open parentheses 1 over straight x close parentheses
therefore space dv over dx equals straight v open square brackets fraction numerator 2 log space straight x over denominator straight x end fraction close square brackets
therefore space dv over dx equals straight x to the power of log space straight x end exponent open parentheses fraction numerator 2 log space straight x over denominator straight x end fraction close parentheses
therefore space dv over dx equals 2 straight x to the power of log space straight x minus 1 end exponent log space straight x space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
From (1), (2), (3), we get,
space space space space dy over dx equals left parenthesis log space straight x right parenthesis to the power of straight x open square brackets fraction numerator 1 over denominator log space straight x end fraction plus log left parenthesis log space straight x right parenthesis close square brackets plus 2 straight x to the power of log space straight x minus 1 end exponent. log space straight x

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