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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

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

Let space straight y equals 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 Put space left parenthesis log space straight x right parenthesis to the power of straight x equals straight u comma space straight x to the power of logx 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... left parenthesis 1 right parenthesis Now space straight u equals left parenthesis log space straight x right parenthesis to the power of straight x therefore space log space straight u equals log left parenthesis log space straight x right parenthesis to the power of straight x therefore space log space straight u equals straight x. log left parenthesis log space straight x right parenthesis therefore space 1 over straight u du 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 therefore space du over dx equals straight u 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 therefore space du over dx equals left parenthesis log space straight x right parenthesis to the power of straight x 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 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 straight x to the power of log space straight x end exponent therefore space log space straight v equals log left parenthesis straight x to the power of log space straight x end exponent right parenthesis space space space space space space space space space space space space space equals log space straight x. log space straight x equals left parenthesis log space straight x right parenthesis squared therefore space 1 over straight v dv over dx equals left parenthesis 2 log space straight x right parenthesis. open parentheses 1 over straight x close parentheses therefore space dv over dx equals straight v open square brackets fraction numerator 2 log space straight x over denominator straight x end fraction close square brackets therefore space dv over dx equals straight x 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 therefore space dv over dx equals 2 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 space space space space... left parenthesis 3 right parenthesis

From (1), (2), (3), we get,

space space space space dy over dx equals left parenthesis log space straight x right parenthesis to the power of straight x 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 plus 2 straight x to the power of log space straight x minus 1 end exponent. log space straight x

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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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