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Continuity And Differentiability

Question
CBSEENMA12035228
Wired Faculty App

If space straight y equals fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction plus log square root of 1 minus straight x squared end root comma space prove space that space dy over dx equals fraction numerator sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

Solution
space space space space space space space space space space space straight y equals fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction plus log left parenthesis 1 minus straight x squared right parenthesis to the power of 1 half end exponent therefore space space space space space space space straight y equals fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction plus 1 half log left parenthesis 1 minus straight x squared right parenthesis therefore space dy over dx equals straight d over dx open parentheses fraction numerator straight x space sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction close parentheses plus 1 half straight d over dx left square bracket log left parenthesis 1 minus straight x squared right parenthesis right square bracket space space space space space space space space space space space space space equals fraction numerator square root of 1 minus straight x squared end root. begin display style straight d over dx end style left parenthesis straight x space sin to the power of negative 1 end exponent straight x right parenthesis minus left parenthesis straight x space sin to the power of negative 1 end exponent straight x right parenthesis. begin display style straight d over dx end style left parenthesis square root of 1 minus straight x squared end root right parenthesis over denominator 1 minus straight x squared end fraction plus 1 half fraction numerator negative 2 straight x over denominator 1 minus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator square root of 1 minus straight x squared end root. open parentheses begin display style fraction numerator straight x over denominator square root of 1 minus straight x squared end root end fraction end style plus sin to the power of negative 1 end exponent straight x.1 close parentheses space minus straight x space sin to the power of negative 1 end exponent straight x. begin display style fraction numerator negative 2 straight x over denominator 2 square root of 1 minus straight x squared end root end fraction end style over denominator 1 minus straight x squared end fraction minus fraction numerator straight x over denominator 1 minus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator straight x plus square root of 1 minus straight x squared end root space sin to the power of negative 1 end exponent straight x plus begin display style fraction numerator straight x squared sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction end style over denominator 1 minus straight x squared end fraction minus fraction numerator straight x over denominator 1 minus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator straight x plus square root of 1 minus straight x squared end root space sin to the power of negative 1 end exponent straight x plus begin display style fraction numerator straight x squared sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction end style minus straight x over denominator 1 minus straight x squared end fraction space space space space space space space space space space space space space equals fraction numerator square root of 1 minus straight x squared end root space sin to the power of negative 1 end exponent straight x plus begin display style fraction numerator straight x squared sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction end style over denominator 1 minus straight x squared end fraction equals fraction numerator left parenthesis 1 minus straight x squared right parenthesis sin to the power of negative 1 end exponent straight x plus straight x squared space sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space space space space space space space space space space space space space equals fraction numerator sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

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