Question

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

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Solution

space space space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent straight x therefore space space space dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space space space space space space space rightwards double arrow space space square root of 1 minus straight x squared end root dy over dx equals 0 therefore space left parenthesis 1 minus straight x squared right parenthesis open parentheses dy over dx close parentheses squared equals 0 Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space left parenthesis 1 minus straight x squared right parenthesis.2 dy over dx fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses squared. left parenthesis negative 2 space straight x right parenthesis equals 0 Dividing space both space sides space by space 2 dy over dx comma space we space get comma space left parenthesis 1 minus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx equals 0

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