Question

$If space straight x equals tan open parentheses 1 over straight a log space straight y close parentheses comma space show space that space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus left parenthesis 2 space straight x minus straight a right parenthesis dy over dx equals 0.$

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Solution

Here space space space straight x equals tan open parentheses 1 over straight a log space straight y close parentheses therefore space tan to the power of negative 1 end exponent straight x equals 1 over straight a log space straight y space space or space space log space straight y equals straight a space tan to the power of negative 1 end exponent straight x Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space space space space space space fraction numerator 1 over denominator space straight y end fraction dy over dx equals fraction numerator straight a over denominator 1 plus straight x squared end fraction or space space space left parenthesis 1 plus straight x squared right parenthesis dy over dx equals straight a space straight y Again space differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space space space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx space 2 space straight x equals straight a dy over dx therefore space space space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus left parenthesis 2 space straight x minus straight a right parenthesis dy over dx equals 0.

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