Question

$If space straight y equals tan to the power of negative 1 end exponent space straight x comma space show space that space open parentheses 1 plus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction plus 2 space straight x dy over dx equals 0$

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Solution

straight y equals tan to the power of negative 1 end exponent space straight x space space space rightwards double arrow space space dy over dx equals fraction numerator 1 over denominator 1 plus straight x squared end fraction space space space rightwards double arrow space left parenthesis 1 plus straight x squared right parenthesis dy over dx equals 0 Again space differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get. 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 2 space straight x dy over dx equals 0

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