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

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

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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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