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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

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

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

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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

Short Answer Type

269. Differentiate the following w.r.t. x : xx + x1/x

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