Question

$If space straight y space equals space cos to the power of negative 1 end exponent space straight x comma space find space fraction numerator straight d squared straight y over denominator dx squared end fraction space in space terms space of space straight y space alone.$

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Solution

space space space space space space space space space space straight y space equals space cos to the power of negative 1 end exponent space straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space dy over dx equals negative fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction equals negative open parentheses 1 minus straight x squared close parentheses to the power of negative 1 half end exponent therefore space dy over dx equals 1 half open parentheses 1 minus straight x squared close parentheses to the power of 3 over 2 end exponent left parenthesis negative 2 straight x right parenthesis equals negative straight x over open parentheses 1 minus straight x squared close parentheses to the power of 3 over 2 end exponent equals negative fraction numerator cos space straight y over denominator open parentheses 1 minus cos squared close parentheses to the power of 3 over 2 end exponent end fraction space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space space space space equals negative fraction numerator cos space straight y over denominator open parentheses 1 minus cos squared close parentheses to the power of 3 over 2 end exponent end fraction equals negative fraction numerator cos space straight y over denominator sin cubed space straight y end fraction equals negative fraction numerator cos space straight y over denominator sin space straight y end fraction. fraction numerator 1 over denominator sin squared space straight y end fraction equals negative cot space straight y space cosec squared space straight y.

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Continuity And Differentiability

$If space straight y space equals space cos to the power of negative 1 end exponent space straight x comma space find space fraction numerator straight d squared straight y over denominator dx squared end fraction space in space terms space of space straight y space alone.$

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