+91-8076753736 support@wiredfaculty.com
logo
logo

Sponsor Area

Continuity And Differentiability

Question
CBSEENMA12034855
Wired Faculty App

Differentiate xlog x + (log x)x w.r.t.x 

Solution
Let space space space space space space space space space straight y equals straight x to the power of log space straight x end exponent plus left parenthesis log space straight x right parenthesis to the power of straight x Put space straight x space to the power of log space straight x end exponent equals straight u left parenthesis log space straight x right parenthesis to the power of straight x equals straight v therefore space space space space space space space space space space space straight y equals straight u plus straight v therefore space space space space 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 space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space straight u equals straight x to the power of log space straight x end exponent 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 rightwards double arrow space space log space straight u equals log left parenthesis straight x to the power of log space straight x end exponent right parenthesis rightwards double arrow space log space straight u equals log space straight x space log space straight x 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 rightwards double arrow space space log space straight u equals left parenthesis log space straight x right parenthesis squared therefore space 1 over straight u du over dx equals fraction numerator 2 log space straight x over denominator straight x end fraction 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 rightwards double arrow space space du over dx equals straight u open parentheses fraction numerator 2 log space straight x over denominator straight x end fraction close parentheses rightwards double arrow space du over dx equals straight x space 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 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 rightwards double arrow space space du over dx equals 2 space 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... left parenthesis 2 right parenthesis Also space straight v equals left parenthesis log space straight x right parenthesis to the power of straight x 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 rightwards double arrow space space space log space straight v equals log left parenthesis log space straight x right parenthesis to the power of straight x therefore space space log space straight v equals straight x. log left parenthesis log space straight x right parenthesis Differentiating space both space sides space straight w. straight r. straight t. straight x comma
space 1 over straight v dv 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 rightwards double arrow space space dv over dx equals straight v 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 rightwards double arrow space dv over dx equals left parenthesis log space straight x right parenthesis to the power of straight x open square brackets fraction numerator 1 plus log. log left parenthesis log space straight x right parenthesis over denominator log space straight x end fraction close square brackets rightwards double arrow space space dv over dx equals left parenthesis log space straight x right parenthesis to the power of straight x minus 1 end exponent left square bracket 1 plus log space straight x. log left parenthesis log space straight x right parenthesis 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... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get comma dy over dx equals 2 straight x to the power of log space straight x minus 1 end exponent space log space straight x plus left parenthesis log space straight x right parenthesis to the power of straight x minus 1 end exponent space left square bracket 1 plus log space straight x. log left parenthesis log space straight x right parenthesis right square bracket

Some More Questions From Continuity and Differentiability Chapter

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation