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Continuity And Differentiability

Question
CBSEENMA12035044

Differentiate cos(xx) w.r.t.x.

Solution
Let space space space space space space space straight y equals cos left parenthesis straight x to the power of straight x right parenthesis
therefore space space space space space space space space space straight y equals cos space straight u space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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
where space straight u equals straight x to the power of straight x
therefore 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 space rightwards double arrow 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... left parenthesis 2 right parenthesis
From space left parenthesis 1 right parenthesis comma space dy over dx equals negative sin space straight u du over dx
therefore space dy over dx equals negative sin left parenthesis straight x to the power of straight x right parenthesis. 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 left square bracket because space of space left parenthesis 2 right parenthesis right square bracket
therefore space dy over dx equals negative straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis. sin left parenthesis straight x to the power of straight x right parenthesis

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