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Continuity And Differentiability

Question

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Differentiate x.logx.log(log x)w.r.t.x.

Solution

Let space space space space space space straight y equals straight x. log space straight x. log left parenthesis log space straight x right parenthesis therefore space log space straight y equals log left square bracket straight x. log space straight x. log left parenthesis log space straight x right parenthesis right square bracket rightwards double arrow space log space straight y equals log space straight x plus log left parenthesis log space straight x right parenthesis plus log left square bracket log left parenthesis log space straight x right parenthesis right square bracket Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get space 1 over straight y dy over dx equals 1 over straight x plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus fraction numerator 1 over denominator log left parenthesis log space straight x right parenthesis end fraction. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x therefore space dy over dx equals straight y open square brackets 1 over straight x plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus fraction numerator 1 over denominator log left parenthesis log space straight x right parenthesis end fraction. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close square brackets or space space dy over dx equals straight x. log space straight x. log left parenthesis log space straight x right parenthesis open square brackets 1 over straight x plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus fraction numerator 1 over denominator log left parenthesis log space straight x right parenthesis end fraction. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close square brackets therefore space space dy over dx equals log space straight x. log left parenthesis log space straight x right parenthesis plus log left parenthesis log space straight x right parenthesis plus 1 therefore space space dy over dx equals 1 plus left parenthesis 1 plus log space straight x right parenthesis. log left parenthesis log space straight x right parenthesis.

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