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Continuity And Differentiability

Question
CBSEENMA12034871
Wired Faculty App

 Differentiate the following w.r.t. x :
xx + xa + ax + aa for some fixed a > 0 and x > 0

Solution
Let space straight y equals straight x to the power of straight x plus straight x to the power of straight a plus straight a to the power of straight x plus straight a to the power of straight a therefore space straight y equals left parenthesis straight x to the power of straight a plus straight a to the power of straight x plus straight a to the power of straight a right parenthesis plus straight x to the power of straight x Put space space straight x to the power of straight a plus straight a to the power of straight x plus straight a to the power of straight a equals straight u comma space straight x to the power of straight x 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 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 1 right parenthesis Also space straight u equals space straight x to the power of straight a plus straight a to the power of straight x plus straight a to the power of straight a therefore space du over dx equals ax to the power of straight a minus 1 end exponent plus straight a to the power of straight x. log space straight a plus 0 therefore space du over dx equals ax to the power of straight a minus 1 end exponent plus straight a to the power of straight x. log space straight a 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 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 straight x therefore space log space straight v equals log space straight x to the power of straight x equals straight x. log space straight x therefore space 1 over straight v dv over dx equals straight x.1 over straight x plus log space straight x.1 therefore space dv over dx equals straight x to the power of straight x left square bracket 1 plus log space straight x right square bracket 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 space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get comma space space space space space dy over dx equals ax to the power of straight a minus 1 end exponent plus straight a to the power of straight x. log space straight a plus straight x to the power of straight x left square bracket 1 plus log space straight x right square bracket.

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