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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $straight x to the power of cos to the power of negative 1 end exponent straight x end exponent$

Solution

space space space space space Let space straight y equals straight x to the power of cos to the power of negative 1 end exponent straight x end exponent therefore space log space straight y equals log space straight x to the power of cos to the power of negative 1 end exponent straight x end exponent equals cos to the power of negative 1 end exponent straight x. log space straight x Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space 1 over straight y dy over dx equals left parenthesis cos to the power of negative 1 end exponent straight x right parenthesis. open parentheses 1 over straight x close parentheses plus left parenthesis log space straight x right parenthesis open parentheses negative fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction close parentheses therefore space dy over dx equals straight y open square brackets fraction numerator cos to the power of negative 1 end exponent straight x over denominator straight x end fraction minus fraction numerator log space straight x over denominator square root of 1 minus straight x squared end root end fraction close square brackets therefore space dy over dx equals space straight x to the power of cos to the power of negative 1 end exponent straight x end exponent open square brackets fraction numerator cos to the power of negative 1 end exponent straight x over denominator straight x end fraction minus fraction numerator log space straight x over denominator square root of 1 minus straight x squared end root end fraction close square brackets

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