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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root$

Solution

Let space straight y equals square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root equals left square bracket log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis right square bracket to the power of begin inline style 1 half end style end exponent therefore space dy over dx equals 1 half left square bracket log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis right square bracket to the power of negative 1 half end exponent straight d over dx left square bracket log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis right square bracket space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root end fraction cross times fraction numerator 1 over denominator straight x cubed plus straight e to the power of straight x end fraction cross times straight d over dx left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root end fraction cross times fraction numerator 1 over denominator straight x cubed plus straight e to the power of straight x end fraction cross times left parenthesis 3 straight x squared plus straight e to the power of straight x right parenthesis equals fraction numerator 3 straight x squared plus straight e to the power of straight x over denominator 2 left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis square root of log space left parenthesis straight x cubed plus straight e to the power of straight x right parenthesis end root end fraction

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