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Continuity And Differentiability

Question
CBSEENMA12034750
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If space straight y equals square root of straight x squared plus 1 end root minus log open parentheses 1 over straight x plus square root of 1 plus 1 over straight x squared end root close parentheses comma space find space dy over dx.

Solution
Here space straight y equals square root of straight x squared plus 1 end root minus log open parentheses 1 over straight x plus square root of 1 plus 1 over straight x squared end root close parentheses therefore space straight y equals square root of straight x squared plus 1 end root minus log open parentheses 1 over straight x plus square root of fraction numerator straight x squared plus 1 over denominator straight x end fraction end root close parentheses or space straight y equals square root of straight x squared plus 1 end root minus log open parentheses fraction numerator 1 plus square root of straight x squared plus 1 end root over denominator straight x end fraction close parentheses therefore straight y equals square root of straight x squared plus 1 end root minus log open parentheses 1 plus square root of straight x squared plus 1 end root close parentheses plus log space straight x therefore space dy over dx equals fraction numerator 1 over denominator 2 square root of straight x squared plus 1 end root end fraction. straight d over dx left parenthesis straight x squared plus 1 right parenthesis minus fraction numerator 1 over denominator 1 plus square root of straight x squared plus 1 end root end fraction. straight d over dx open parentheses 1 plus square root of straight x squared plus 1 end root close parentheses plus 1 over straight x space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of straight x squared plus 1 end root end fraction.2 straight x minus fraction numerator 1 over denominator 1 plus square root of straight x squared plus 1 end root end fraction. open parentheses 0 plus fraction numerator 2 straight x over denominator 2 square root of straight x squared plus 1 end root end fraction close parentheses plus 1 over straight x space space space space space space space space space space space space space equals fraction numerator straight x over denominator square root of straight x squared plus 1 end root end fraction minus fraction numerator straight x over denominator square root of straight x squared plus 1 end root open parentheses 1 plus square root of straight x squared plus 1 end root close parentheses end fraction plus 1 over straight x equals fraction numerator straight x over denominator square root of straight x squared plus 1 end root end fraction open square brackets 1 minus fraction numerator 1 over denominator 1 plus square root of straight x squared plus 1 end root end fraction close square brackets plus 1 over straight x equals fraction numerator straight x over denominator square root of straight x squared plus 1 end root end fraction open square brackets fraction numerator 1 plus square root of straight x squared plus 1 end root minus 1 over denominator 1 plus square root of straight x squared plus 1 end root end fraction close square brackets plus 1 over straight x equals fraction numerator straight x over denominator square root of straight x squared plus 1 end root end fraction cross times fraction numerator square root of straight x squared plus 1 end root over denominator 1 plus square root of straight x squared plus 1 end root end fraction plus 1 over straight x equals fraction numerator straight x over denominator 1 plus square root of straight x squared plus 1 end root end fraction plus 1 over straight x

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