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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. $If space straight x to the power of straight p. straight y to the power of straight q equals left parenthesis straight x plus straight y right parenthesis to the power of straight p plus straight q end exponent comma space show space that space dy over dx equals straight y over straight x$

space space space space 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 rightwards double arrow space log left parenthesis straight x to the power of straight p. straight y to the power of straight q right parenthesis equals log left parenthesis straight x plus straight y right parenthesis to the power of straight p plus straight q end exponent rightwards double arrow space log space straight x to the power of straight p plus log space straight y to the power of straight q equals log left parenthesis straight x plus straight y right parenthesis to the power of straight p plus straight q end exponent rightwards double arrow straight p space log space straight x plus straight q space log space straight y equals left parenthesis straight p plus straight q right parenthesis log left parenthesis straight x plus straight y right parenthesis Differentiating space straight w. straight r. straight t. straight x comma space we space get comma space space space space space straight p over straight x plus straight q over straight y dy over dx equals fraction numerator straight p plus straight q over denominator straight x plus straight y end fraction open square brackets 1 plus dy over dx close square brackets therefore space straight p over straight x plus straight q over straight y dy over dx equals fraction numerator straight p plus straight q over denominator straight x plus straight y end fraction plus fraction numerator straight p plus straight q over denominator straight x plus straight y end fraction dy over dx space space space rightwards double arrow space space open parentheses straight q over straight y minus fraction numerator straight p plus straight q over denominator straight x plus straight y end fraction close parentheses dy over dx equals fraction numerator straight p plus straight q over denominator straight x plus straight y end fraction minus straight p over straight x therefore space open square brackets fraction numerator qz plus qy minus py minus qy over denominator straight y left parenthesis straight x plus straight y right parenthesis end fraction close square brackets dy over dx equals fraction numerator px plus qx minus px minus py over denominator straight x left parenthesis straight x plus straight y right parenthesis end fraction therefore space open square brackets fraction numerator qx minus py over denominator straight y end fraction close square brackets dy over dx equals fraction numerator qx minus py over denominator straight x end fraction space rightwards double arrow space space 1 over straight y dy over dx equals 1 over straight x therefore 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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