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Continuity And Differentiability

Question

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Differentiate $square root of a italic plus square root of a italic plus square root of a italic plus x to the power of italic 2 end root end root end root$ w.r.t x

Solution

Let space straight y equals square root of straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root end root equals open square brackets straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root close square brackets to the power of begin inline style 1 half end style end exponent therefore space dy over dx equals 1 half open square brackets straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root close square brackets to the power of negative 1 half end exponent straight d over dx open square brackets straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root close square brackets equals fraction numerator 1 over denominator 2 square root of straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root end root end fraction cross times straight d over dx open parentheses straight a plus square root of straight a plus straight x squared end root close parentheses to the power of begin inline style 1 half end style end exponent equals fraction numerator 1 over denominator 2 square root of straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root end root end fraction cross times 1 half open parentheses straight a plus square root of straight a plus straight x squared end root close parentheses to the power of begin inline style negative 1 half end style end exponent. straight d over dx left parenthesis straight a plus straight x squared right parenthesis to the power of begin inline style 1 half end style end exponent equals fraction numerator 1 over denominator 2 square root of straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root end root end fraction cross times fraction numerator 1 over denominator 2 square root of straight a plus square root of straight a plus straight x squared end root end root end fraction cross times 1 half left parenthesis straight a plus straight x squared right parenthesis to the power of begin inline style negative 1 half end style end exponent straight d over dx left parenthesis straight a plus straight x squared right parenthesis equals fraction numerator 1 over denominator 2 square root of straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root end root end fraction cross times fraction numerator 1 over denominator 2 square root of straight a plus square root of straight a plus straight x squared end root end root end fraction cross times fraction numerator 1 over denominator 2 square root of straight a plus straight x squared end root end fraction cross times 2 straight x equals fraction numerator 1 over denominator 4 square root of straight a plus straight x squared end root square root of straight a plus square root of straight a plus straight x squared end root end root square root of straight a plus square root of straight a plus square root of straight a plus straight x squared end root end root end root end fraction

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