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Continuity And Differentiability

Question
CBSEENMA12035242
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If space straight y equals sin to the power of negative 1 end exponent left parenthesis straight x squared square root of 1 minus straight x squared end root plus straight x square root of 1 minus straight x to the power of 4 end root right parenthesis comma space show space that space dy over dx minus fraction numerator 2 straight x over denominator square root of 1 minus straight x to the power of 4 end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x to the power of 4 end root end fraction

Solution
space straight y equals sin to the power of negative 1 end exponent left parenthesis straight x squared square root of 1 minus straight x squared end root plus straight x square root of 1 minus straight x to the power of 4 end root right parenthesis Put space straight x squared equals sin space straight theta comma space straight x equals sin space straight ϕ therefore space straight y equals sin to the power of negative 1 end exponent left parenthesis sin space straight theta square root of 1 minus sin squared space straight ϕ end root plus sin space straight ϕ square root of 1 minus sin squared space straight theta end root right parenthesis space space space space space space space equals sin to the power of negative 1 end exponent left parenthesis sin space straight theta space cos space straight ϕ plus cos space straight theta space sin space straight ϕ right parenthesis equals sin to the power of negative 1 end exponent left square bracket sin left parenthesis straight theta plus straight ϕ right parenthesis right square bracket therefore space straight y equals straight theta plus straight ϕ space space rightwards double arrow space straight y equals sin to the power of negative 1 end exponent space straight x squared plus sin to the power of negative 1 end exponent space straight x dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x to the power of 4 end root end fraction 2 straight x plus fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space rightwards double arrow space dy over dx minus fraction numerator 2 straight x over denominator square root of 1 minus straight x to the power of 4 end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction

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