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Continuity and Differentiability

Short Answer Type

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

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262.

If space straight x to the power of straight p. straight y to the power of straight q equals left parenthesis straight x plus straight y right parenthesis to the power of straight p plus straight q end exponent comma space show space that space dy over dx equals straight y over straight x

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Long Answer Type

263. Differentiate x^{log x} + (log x)^x w.r.t.x

Let space space space space space space space space space straight y equals straight x to the power of log space straight x end exponent plus left parenthesis log space straight x right parenthesis to the power of straight x Put space straight x space to the power of log space straight x end exponent equals straight u left parenthesis log space straight x right parenthesis to the power of straight x equals straight v therefore space space space space space space space space space space space straight y equals straight u plus straight v therefore space space space space 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 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 straight x to the power of log space straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space log space straight u equals log left parenthesis straight x to the power of log space straight x end exponent right parenthesis rightwards double arrow space log space straight u equals log space straight x space log space straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space log space straight u equals left parenthesis log space straight x right parenthesis squared therefore space 1 over straight u du over dx equals fraction numerator 2 log space straight x over denominator 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 rightwards double arrow space space du over dx equals straight u open parentheses fraction numerator 2 log space straight x over denominator straight x end fraction close parentheses rightwards double arrow space du over dx equals straight x space to the power of log space straight x end exponent open parentheses fraction numerator 2 log space straight x over denominator straight x end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space du over dx equals 2 space straight x to the power of log space straight x minus 1 end exponent log space straight x space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Also space straight v equals left parenthesis log space straight x right parenthesis to the power of straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space log space straight v equals log left parenthesis log space straight x right parenthesis to the power of straight x therefore space space log space straight v equals straight x. log left parenthesis log space straight x right parenthesis Differentiating space both space sides space straight w. straight r. straight t. straight x comma

space 1 over straight v dv over dx equals straight x. open parentheses fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close parentheses plus log left parenthesis log space straight x right parenthesis.1 rightwards double arrow space space dv over dx equals straight v open square brackets fraction numerator 1 over denominator log space straight x end fraction plus log left parenthesis log space straight x right parenthesis close square brackets space rightwards double arrow space dv over dx equals left parenthesis log space straight x right parenthesis to the power of straight x open square brackets fraction numerator 1 plus log. log left parenthesis log space straight x right parenthesis over denominator log space straight x end fraction close square brackets rightwards double arrow space space dv over dx equals left parenthesis log space straight x right parenthesis to the power of straight x minus 1 end exponent left square bracket 1 plus log space straight x. log left parenthesis log 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... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get comma dy over dx equals 2 straight x to the power of log space straight x minus 1 end exponent space log space straight x plus left parenthesis log space straight x right parenthesis to the power of straight x minus 1 end exponent space left square bracket 1 plus log space straight x. log left parenthesis log space straight x right parenthesis right square bracket

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Short Answer Type

264. Differentiate the following functions w.r.t. x :

straight x to the power of straight x plus straight x to the power of log space straight x end exponent

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265. Differentiate the following functions w.r.t. x :

left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log space straight x end exponent

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266. Differentiate the following functions w.r.t. x :

open parentheses straight x plus 1 over straight x close parentheses to the power of straight x plus straight x to the power of open parentheses straight x plus 1 over straight x close parentheses end exponent

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267. Differentiate the following functions w.r.t. x :

straight x to the power of straight x to the power of 2 minus 3 end exponent end exponent plus left parenthesis straight x minus 3 right parenthesis to the power of straight x squared end exponent space for space straight x greater than 3

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268. Differentiate the following w.r.t. x :(log x)^{log x} + (1 + x)^2x

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269. Differentiate the following w.r.t. x :
x^x + x^1/x

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270.

Differentiate the following w.r.t. x
x^x + (1 + x)^{log x}

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Previous Year Papers

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Short Answer Type

Long Answer Type

263. Differentiate xlog x + (log x)x w.r.t.x

Short Answer Type

263. Differentiate x^{log x} + (log x)^x w.r.t.x