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Continuity And Differentiability

Question
CBSEENMA12034715
Wired Faculty App

If space straight y equals open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n comma space prove space that space dy over dn equals fraction numerator ny over denominator square root of straight x squared plus straight a squared end root end fraction.

Solution
Here space straight y equals open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space space space space space space space space space space space Differentiating space both space sides space straight w. straight r. straight t. space straight x space we space get comma dy over dx equals straight n open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n minus 1 end exponent. straight d over dx open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses space space space space space space space space equals straight n open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n minus 1 end exponent. open parentheses 1 plus fraction numerator 2 straight x over denominator 2 square root of straight x squared plus straight a squared end root end fraction close parentheses space space space space space space space space equals straight n open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n minus 1 end exponent. open parentheses 1 plus fraction numerator straight x over denominator square root of straight x squared plus straight a squared end root end fraction close parentheses space space space space space space space space equals straight n open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n minus 1 end exponent. open parentheses fraction numerator square root of straight x squared plus straight a squared end root plus straight x over denominator square root of straight x squared plus straight a squared end root end fraction close parentheses equals fraction numerator straight n open parentheses straight x plus square root of straight x squared plus straight a squared end root close parentheses to the power of straight n space over denominator square root of straight x squared plus straight a squared end root end fraction therefore dy over dx equals fraction numerator ny over denominator square root of straight x squared plus straight a squared end root end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space 1 right square bracket

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