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Continuity And Differentiability

Question
CBSEENMA12035324
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Find the derivative w.r.t.x of the function 
left parenthesis log subscript cos space straight x end subscript sin space straight x right parenthesis space left parenthesis log subscript sin space straight x end subscript space cos space straight x right parenthesis to the power of negative 1 end exponent plus sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses space at space straight x equals straight pi over 4

Solution
Let space straight y equals left parenthesis log subscript cos space straight x end subscript sin space straight x right parenthesis space left parenthesis log subscript sin space straight x end subscript space cos space straight x right parenthesis to the power of negative 1 end exponent plus sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses space Put space straight u equals left parenthesis log subscript cos space straight x end subscript sin space straight x right parenthesis space left parenthesis log subscript sin space straight x end subscript space cos space straight x right parenthesis to the power of negative 1 end exponent comma space straight v equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses space therefore space space space straight y equals straight u plus straight v rightwards double arrow 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... left parenthesis 1 right parenthesis Now space straight u equals log subscript cos space straight x end subscript sin space straight x. left parenthesis log subscript sin space straight x end subscript space cos space straight x right parenthesis to the power of negative 1 end exponent therefore space straight u equals fraction numerator log space sin space straight x over denominator log space cos space straight x end fraction. open parentheses fraction numerator log space cos space straight x over denominator log space sin space straight x end fraction close parentheses to the power of negative 1 end exponent equals fraction numerator log space sin space straight x over denominator log space cos space straight x end fraction cross times fraction numerator log space sin space straight x over denominator log space cos space straight x end fraction therefore space straight u equals open parentheses fraction numerator log space sin space straight x over denominator log space cos space straight x end fraction close parentheses squared therefore space du over dx equals 2 fraction numerator log space sin space straight x over denominator log space cos space straight x end fraction cross times fraction numerator left parenthesis log space cos space straight x right parenthesis. begin display style fraction numerator cos space straight x over denominator sin space straight x end fraction end style minus left parenthesis log space sin space straight x right parenthesis. begin display style fraction numerator negative sin space straight x over denominator cos space straight x end fraction end style over denominator left parenthesis log space cos space straight x right parenthesis squared end fraction therefore space du over dx equals fraction numerator 2 left parenthesis log space sin space straight x right parenthesis over denominator log space cos space straight x end fraction cross times fraction numerator cot space straight x space left parenthesis log space cos space straight x right parenthesis plus tan space straight x space left parenthesis log space sin space straight x right parenthesis over denominator left parenthesis log space cos space straight x right parenthesis squared end fraction rightwards double arrow space du over dx equals fraction numerator 2 left parenthesis log space sin space straight x right parenthesis left square bracket cot space straight x left parenthesis log space cos space straight x right parenthesis plus tan space straight x left parenthesis log space sin space straight x right parenthesis right square bracket over denominator left parenthesis log space cos space straight x right parenthesis cubed 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... left parenthesis 2 right parenthesis Also space straight v equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses semicolon space Put space straight x equals tan space straight theta therefore space straight v equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 tan space straight theta over denominator 1 plus tan squared space straight theta end fraction close parentheses equals sin to the power of negative 1 end exponent left parenthesis sin space 2 straight theta right parenthesis equals 2 straight theta equals 2 space tan to the power of negative 1 end exponent straight x therefore space dv over dx equals fraction numerator 2 over denominator 1 plus straight x squared 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 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 (i), (2), (3), we get,
space dy over dx equals fraction numerator 2 left parenthesis log space sin space straight x right parenthesis left square bracket cot space straight x left parenthesis log space cos space straight x right parenthesis plus tan space straight x left parenthesis log space sin space straight x right parenthesis right square bracket over denominator left parenthesis log space cos space straight x right parenthesis cubed end fraction plus fraction numerator 2 over denominator 1 plus straight x squared end fraction At space space space straight x equals straight pi over 4 comma space we space get comma space space dy over dx equals fraction numerator 2 open parentheses log space sin space begin display style straight pi over 4 end style close parentheses open square brackets cot space begin display style straight pi over 4 end style open parentheses log space cos space begin display style straight pi over 4 end style close parentheses plus tan space begin display style straight pi over 4 end style open parentheses log space sin space begin display style straight pi over 4 end style close parentheses close square brackets over denominator open parentheses log space cos space begin display style straight pi over 4 end style close parentheses cubed end fraction plus fraction numerator 2 over denominator 1 plus open parentheses begin display style straight pi over 4 end style close parentheses squared end fraction space space space space space space space space space space equals fraction numerator 2 open parentheses log begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses open square brackets 1 open parentheses log space begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses plus 1. open parentheses log space begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses close square brackets over denominator open parentheses log space begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses cubed end fraction plus fraction numerator 2 over denominator 1 plus begin display style straight pi squared over 16 end style end fraction space space space space space space space space space space equals fraction numerator 4 open parentheses log begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses squared over denominator open parentheses log begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses end fraction plus fraction numerator 32 over denominator straight pi squared plus 16 end fraction equals fraction numerator 4 over denominator log space begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction plus fraction numerator 32 over denominator straight pi squared plus 16 end fraction space space space space space space space space space space equals fraction numerator 4 over denominator log space 1 minus log space 2 to the power of begin display style 1 half end style end exponent end fraction plus fraction numerator 32 over denominator straight pi squared plus 16 end fraction equals fraction numerator 4 over denominator negative begin display style 1 half end style log space 2 end fraction plus fraction numerator 32 over denominator straight pi squared plus 16 end fraction equals negative fraction numerator 8 over denominator log space 2 end fraction plus fraction numerator 32 over denominator straight pi squared plus 16 end fraction

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