Differentiate : w.r.t x.

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Question

Differentiate : $5 to the power of square root of straight x squared plus 1 end root end exponent plus open parentheses square root of straight x squared plus 1 end root close parentheses to the power of 5$ w.r.t x.

Solution

Let space straight y equals 5 to the power of square root of straight x squared plus 1 end root end exponent plus open parentheses square root of straight x squared plus 1 end root close parentheses to the power of 5 equals 5 to the power of square root of straight x squared plus 1 end root end exponent plus left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 5 over 2 end style end exponent therefore space dy over dx equals 5 to the power of square root of straight x squared plus 1 end root end exponent. log space 5. straight d over dx left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 1 half end style end exponent plus 5 over 2 left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 3 over 2 end style end exponent. straight d over dx left parenthesis straight x squared plus 1 right parenthesis space space space space space space space space space space space space space equals 5 to the power of square root of straight x squared plus 1 end root end exponent. log space 5.1 half left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style negative 1 half end style end exponent.2 straight x plus 5 over 2 left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 3 over 2 end style end exponent.2 straight x space space space space space space space space space space space space space equals fraction numerator 5 to the power of square root of straight x squared plus 1 end root end exponent. log space 5 over denominator square root of straight x squared plus 1 end root end fraction plus 5 straight x left parenthesis straight x squared plus 1 right parenthesis to the power of begin inline style 3 over 2 end style end exponent.

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Continuity And Differentiability

Differentiate : $5 to the power of square root of straight x squared plus 1 end root end exponent plus open parentheses square root of straight x squared plus 1 end root close parentheses to the power of 5$ w.r.t x.

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