Question

$straight y equals sin to the power of negative 1 end exponent open curly brackets straight x square root of 1 minus straight x end root minus square root of straight x square root of 1 minus straight x squared end root close curly brackets space semicolon space find space dy over dx comma$

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Solution

space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent open curly brackets straight x square root of 1 minus straight x end root minus square root of straight x square root of 1 minus straight x squared end root close curly brackets Put space space space space straight x equals sin space straight theta comma space square root of straight x equals sin space straight ϕ therefore space space space space space space straight y equals sin to the power of negative 1 end exponent open curly brackets sin space straight theta square root of 1 minus sin squared space straight ϕ end root minus sin space straight ϕ square root of 1 minus sin squared space straight theta end root close curly brackets space space space space space space space space space space space space equals sin to the power of negative 1 end exponent left curly bracket sin space straight theta space cos space straight ϕ minus sin space straight ϕ space cos space straight theta right curly bracket equals sin to the power of negative 1 end exponent left curly bracket sin left parenthesis straight theta minus straight ϕ right parenthesis right curly bracket equals straight theta minus straight ϕ therefore space space space space space space straight y equals sin to the power of negative 1 end exponent straight x minus sin to the power of negative 1 end exponent square root of straight x therefore space dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction minus fraction numerator 1 over denominator 1 minus straight x end fraction. fraction numerator 1 over denominator 2 square root of straight x end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction minus fraction numerator 1 over denominator 2 square root of straight x square root of 1 minus straight x end root end fraction

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Continuity And Differentiability

$straight y equals sin to the power of negative 1 end exponent open curly brackets straight x square root of 1 minus straight x end root minus square root of straight x square root of 1 minus straight x squared end root close curly brackets space semicolon space find space dy over dx comma$

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