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Continuity And Differentiability

Question
CBSEENMA12035073
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Differentiate the following functions w.r.t. x :left parenthesis sin space straight x right parenthesis to the power of straight x plus sin to the power of negative 1 end exponent square root of straight x

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

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