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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

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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

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

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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

Test Series

Test Yourself

Short Answer Type

Long Answer Type

Short Answer Type

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

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