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Continuity and Differentiability

Short Answer Type

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

Let space space space space space space straight y equals straight x. log space straight x. log left parenthesis log space straight x right parenthesis therefore space log space straight y equals log left square bracket straight x. log space straight x. log left parenthesis log space straight x right parenthesis right square bracket rightwards double arrow space log space straight y equals log space straight x plus log left parenthesis log space straight x right parenthesis plus log left square bracket log left parenthesis log space straight x right parenthesis right square bracket Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get space 1 over straight y dy over dx equals 1 over straight x plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus fraction numerator 1 over denominator log left parenthesis log space straight x right parenthesis end fraction. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x therefore space dy over dx equals straight y open square brackets 1 over straight x plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus fraction numerator 1 over denominator log left parenthesis log space straight x right parenthesis end fraction. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close square brackets or space space dy over dx equals straight x. log space straight x. log left parenthesis log space straight x right parenthesis open square brackets 1 over straight x plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x plus fraction numerator 1 over denominator log left parenthesis log space straight x right parenthesis end fraction. fraction numerator 1 over denominator log space straight x end fraction.1 over straight x close square brackets therefore space space dy over dx equals log space straight x. log left parenthesis log space straight x right parenthesis plus log left parenthesis log space straight x right parenthesis plus 1 therefore space space dy over dx equals 1 plus left parenthesis 1 plus log space straight x right parenthesis. log left parenthesis log space straight x right parenthesis.

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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

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270.

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

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