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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $straight x to the power of log space straight x end exponent$

Solution

Let space straight x to the power of log space straight x end exponent because space log space straight y equals log space straight x to the power of log space straight x end exponent equals log space straight x. log space straight x equals left parenthesis log space straight x right parenthesis squared Differentiating space straight w. straight r. straight t. space straight x comma 1 over straight y dy over dx equals 2 log space straight x. straight d over dx left parenthesis log space straight x right parenthesis space space space space space rightwards double arrow 1 over straight y dy over dx equals 2 log space straight x.1 over straight x therefore space dy over dx equals 2 ylog space straight x.1 over straight x equals 2 straight x to the power of log space straight x end exponent fraction numerator log space straight x over denominator straight x end fraction equals 2 straight x to the power of log space straight x minus 1 end exponent. log space straight x.

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