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Continuity And Differentiability

Question
CBSEENMA12035134
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If space straight y equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction tan to the power of negative 1 end exponent open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses comma space prove space that space dy over dx equals fraction numerator 1 over denominator straight a plus straight b space cos space straight x end fraction comma space straight a greater than straight b greater than 0.

Solution
Here space space straight y equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction tan to the power of negative 1 end exponent open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses therefore space dy over dx equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction. fraction numerator 1 over denominator 1 plus begin display style fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end style tan squared begin display style straight x over 2 end style end fraction. straight d over dx open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses space space space space space space space space space space space space space equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction. fraction numerator straight a plus straight b over denominator straight a plus straight b plus left parenthesis straight a minus straight b right parenthesis space tan squared begin display style straight x over 2 end style end fraction. square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root. sec squared straight x over 2.1 half space space space space space space space space space space space space space equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction cross times fraction numerator straight a plus straight b over denominator 2 end fraction cross times fraction numerator square root of straight a minus straight b end root over denominator square root of straight a plus straight b end root end fraction. fraction numerator sec squared straight x over 2 over denominator straight a open parentheses 1 plus tan squared straight x over 2 close parentheses plus straight b open parentheses 1 minus tan squared straight x over 2 close parentheses end fraction space space space space space space space space space space space space space equals fraction numerator 1 over denominator straight a plus straight b end fraction cross times fraction numerator straight a plus straight b over denominator 1 end fraction. fraction numerator begin display style fraction numerator sec squared straight x over 2 over denominator open parentheses 1 plus tan squared straight x over 2 close parentheses end fraction end style over denominator straight a plus straight b open parentheses begin display style fraction numerator 1 minus tan squared straight x over 2 over denominator 1 plus tan squared straight x over 2 end fraction end style close parentheses end fraction equals fraction numerator begin display style fraction numerator sec squared straight x over 2 over denominator sec squared straight x over 2 end fraction end style over denominator straight a plus straight b space cos space straight x end fraction equals fraction numerator 1 over denominator straight a plus straight b space cos space straight x end fraction

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