Question

$Differentiate space sin to the power of negative 1 end exponent open parentheses fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction close parentheses space straight w. straight r. straight t. straight x comma space using space chain space rule$

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Solution

Let space space space space space space space space space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction close parentheses Put space fraction numerator straight a plus straight b space cos space straight x over denominator straight b plus straight a space cos space straight x end fraction equals straight u therefore space space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent therefore space space space space space dy over du equals fraction numerator 1 over denominator square root of 1 minus straight u squared end root end fraction Also space du over dx equals fraction numerator left parenthesis straight b plus straight a space cos space straight x right parenthesis left parenthesis negative straight b space sin space straight x right parenthesis minus left parenthesis straight a plus straight b space cos space straight x right parenthesis left parenthesis negative straight a space sin space straight x right parenthesis over denominator left parenthesis straight b plus straight a space cos space straight x right parenthesis squared end fraction space space space space space space space space space space space space space space space space space equals fraction numerator negative straight b squared sin space straight x minus absin space straight x space cos space straight x plus straight a squared sin space straight x space plus absin space straight x space cos space straight x over denominator left parenthesis straight b plus straight a space cos space straight x right parenthesis squared end fraction therefore space space space space space du over dx equals fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus straight a space cos space straight x right parenthesis squared end fraction Now space by space chain space rule comma

dy over dx equals dy over du. du over dx equals fraction numerator 1 over denominator square root of 1 minus straight u squared end root end fraction. fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus acos space straight x right parenthesis end fraction space left square bracket because space of left parenthesis 1 right parenthesis comma left parenthesis 2 right parenthesis right square bracket space space space space space space space space equals fraction numerator 1 over denominator square root of 1 minus begin display style fraction numerator left parenthesis straight a plus straight b space cos space straight x right parenthesis squared over denominator left parenthesis straight b plus acos space straight x right parenthesis squared end fraction end style end root end fraction. fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus acos space straight x right parenthesis squared end fraction space space space space space space space space equals fraction numerator straight b plus acos space straight x over denominator square root of left parenthesis straight b plus acos space straight x right parenthesis squared minus left parenthesis straight a plus straight b space cos space straight x right parenthesis squared end root end fraction cross times fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator left parenthesis straight b plus acos space straight x right parenthesis squared end fraction space space space space space space space space equals fraction numerator 1 over denominator square root of left parenthesis straight b squared minus straight a squared right parenthesis minus left parenthesis straight b squared minus straight a squared right parenthesis cos squared straight x end root end fraction cross times fraction numerator left parenthesis straight a squared minus straight b squared right parenthesis sin space straight x over denominator straight b plus acos space straight x end fraction space space space space space space space space equals fraction numerator 1 over denominator square root of straight b squared minus straight a squared end root cross times sin space straight x end fraction cross times fraction numerator negative left parenthesis straight b squared minus straight a squared right parenthesis sin space straight x over denominator straight b plus acos space straight x end fraction equals negative fraction numerator square root of straight b squared minus straight a squared end root over denominator straight b plus acos space straight x end fraction

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