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Continuity And Differentiability

Question

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If y=(log x)², prove that x² $fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx minus 2 equals 0$

Solution

space space space space space space space space space space space space straight y equals left parenthesis log space straight x right parenthesis squared therefore space space space dy over dx equals 2. log space straight x cross times 1 over straight x equals 2 fraction numerator log space straight x over denominator straight x end fraction therefore space straight x dy over dx equals 2 space log space straight x Diffe. space again space straight w. straight r. straight t. straight x comma straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 equals 2 over straight x rightwards double arrow space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals 2 space space rightwards double arrow space straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx minus 2 equals 0

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