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Continuity And Differentiability

Question

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$Differentiate space straight x to the power of straight x sin to the power of negative 1 end exponent square root of straight x space straight w. straight r. straight t. straight x.$

Solution

Let space space space space space straight y equals straight x to the power of straight x sin to the power of negative 1 end exponent square root of straight x Put space space space straight x to the power of straight x equals straight u comma space sin to the power of negative 1 end exponent square root of straight x equals straight v therefore space space space space space space space straight y equals vu therefore space dy over dx equals straight u dv over dx plus straight v dv over dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space space space space space space space straight u equals straight x to the power of straight x therefore space space space space space log space straight u equals log space straight x to the power of straight x space rightwards double arrow space log space straight u equals straight x space log space straight x therefore space 1 over straight u du over dx equals straight x.1 over straight x plus log space straight x.1 rightwards double arrow space du over dx equals straight u left parenthesis 1 plus log space straight x right parenthesis therefore space space du over dx equals straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Also space space straight v equals sin to the power of negative 1 end exponent square root of straight x therefore space dv over dx equals fraction numerator 1 over denominator square root of 1 minus open parentheses square root of straight x close parentheses squared end root end fraction straight d over dx open parentheses square root of straight x close parentheses equals fraction numerator 1 over denominator square root of 1 minus straight x end root end fraction. fraction numerator 1 over denominator 2 square root of straight x end fraction therefore space dv over dx equals fraction numerator 1 over denominator 2 square root of 1 minus straight x end root square root of straight x end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis comma space we space get comma dy over dx equals straight x to the power of straight x. fraction numerator 1 over denominator 2 square root of 1 minus straight x end root square root of straight x end fraction plus sin to the power of negative 1 end exponent square root of straight x. space straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis space space space space space space space space equals fraction numerator straight x to the power of straight x over denominator 2 square root of straight x minus straight x squared end root end fraction plus straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis. space sin to the power of negative 1 end exponent square root of straight x

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