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Continuity And Differentiability

Question
CBSEENMA12034720
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Differentiatefraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root plus square root of bold a bold minus bold x end root end fraction w.r.t x.

Solution
Let space straight y equals fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root plus square root of bold a bold minus bold x end root end fraction equals fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root plus square root of bold a bold minus bold x end root end fraction cross times fraction numerator square root of straight a plus straight x end root minus square root of straight a minus straight x end root over denominator square root of bold a bold plus bold x end root minus square root of bold a bold minus bold x end root end fraction equals fraction numerator left parenthesis straight a plus straight x right parenthesis plus left parenthesis straight a minus straight x right parenthesis minus 2 square root of straight a plus straight x end root square root of straight a minus straight x end root over denominator left parenthesis straight a plus straight x right parenthesis minus left parenthesis straight a minus straight x right parenthesis end fraction equals fraction numerator 2 straight a minus 2 square root of straight a squared minus straight x squared end root over denominator 2 straight x end fraction equals fraction numerator straight a minus square root of straight a squared minus straight x squared end root over denominator straight x end fraction therefore space dy over dx equals fraction numerator straight x begin display style straight d over dx end style open parentheses straight a minus square root of straight a squared minus straight x squared end root close parentheses minus open parentheses straight a minus square root of straight a squared minus straight x squared end root close parentheses begin display style straight d over dx end style left parenthesis straight x right parenthesis over denominator straight x squared end fraction equals fraction numerator straight x open parentheses 0 minus begin display style fraction numerator negative 2 straight x over denominator 2 square root of straight a squared minus straight x squared end root end fraction end style close parentheses minus open parentheses straight a minus square root of straight a squared minus straight x squared end root close parentheses.1 over denominator straight x squared end fraction equals fraction numerator begin display style fraction numerator straight x squared over denominator square root of straight a squared minus straight x squared end root end fraction end style minus straight a plus square root of straight a squared minus straight x squared end root over denominator straight x squared end fraction equals fraction numerator straight x squared minus straight a square root of straight a squared minus straight x squared end root plus straight a squared minus straight x squared over denominator straight x squared square root of straight a squared minus straight x squared end root end fraction equals fraction numerator straight a squared minus straight a square root of straight a squared minus straight x squared end root over denominator straight x squared square root of straight a squared minus straight x squared end root end fraction

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