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Continuity And Differentiability

Question
CBSEENMA12035345
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Find space fraction numerator straight d cubed straight y over denominator dx cubed end fraction space when space straight y equals fraction numerator log space straight x over denominator straight x end fraction.

Solution
straight y equals fraction numerator log space straight x over denominator straight x end fraction Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space dy over dx equals fraction numerator straight x. begin display style straight d over dx end style left parenthesis log space straight x right parenthesis minus log space straight x. begin display style straight d over dx end style left parenthesis straight x right parenthesis over denominator straight x squared end fraction equals fraction numerator straight x. begin display style 1 over straight x end style minus log space straight x.1 over denominator straight x squared end fraction therefore space dy over dx equals fraction numerator 1 minus log space straight x over denominator straight x squared end fraction Differentiating space straight w. straight r. straight t. straight x comma space we space get comma fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator straight x squared begin display style straight d over dx end style left parenthesis 1 minus log space straight x right parenthesis minus left parenthesis 1 minus log space straight x right parenthesis. begin display style straight d over dx end style left parenthesis straight x squared right parenthesis over denominator open parentheses straight x squared close parentheses squared end fraction space space space space space space space space space equals fraction numerator straight x squared begin display style open parentheses negative 1 over straight x close parentheses end style minus left parenthesis 1 minus log space straight x right parenthesis.2 straight x over denominator straight x to the power of 4 end fraction equals fraction numerator negative straight x minus 2 straight x plus 2 straight x space log space straight x over denominator straight x to the power of 4 end fraction space space space space space space space space space equals fraction numerator negative 3 straight x plus 2 xlog space straight x over denominator straight x to the power of 4 end fraction equals fraction numerator negative 3 plus 2 space log space straight x over denominator straight x cubed end fraction fraction numerator straight d cubed straight y over denominator dx cubed end fraction equals fraction numerator straight x cubed open parentheses begin display style 2 over straight x end style close parentheses minus left parenthesis negative 3 plus 2 log space straight x right parenthesis.3 straight x squared over denominator open parentheses straight x cubed close parentheses squared end fraction equals fraction numerator 2 plus 9 minus 6 log space straight x over denominator straight x to the power of 4 end fraction equals fraction numerator 11 minus 6 log space straight x over denominator straight x to the power of 4 end fraction

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