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Continuity and Differentiability

Short Answer Type

241. Differentiate the following w.r.t.x:

left parenthesis straight a plus 2 straight x right parenthesis to the power of 7 straight x end exponent

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242. Differentiate the following w.r.t.x:

left parenthesis 2 straight x plus 3 right parenthesis to the power of straight x minus 5 end exponent

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243. Differentiate the following w.r.t.x:

straight x to the power of log space straight x end exponent

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244. Differentiate the following w.r.t.x:

left parenthesis straight x to the power of straight x right parenthesis to the power of square root of straight x end exponent

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245. Differentiate the following w.r.t.x: $straight x to the power of straight x to the power of straight x end exponent space space space or space straight x to the power of left parenthesis straight x to the power of straight x right parenthesis end exponent$

Let space straight y equals straight x to the power of straight x to the power of straight x end exponent therefore space log space straight y equals log space straight x to the power of straight x to the power of straight x end exponent equals straight x to the power of straight x log space straight x space space space space space space space space space space space space space space space space rightwards double arrow space log left parenthesis log space straight y right parenthesis equals log left square bracket straight x to the power of straight x. log space straight x right square bracket therefore space log left parenthesis log space straight y right parenthesis equals log space straight x to the power of straight x plus log left parenthesis log space straight x right parenthesis space space space space space rightwards double arrow space log left parenthesis log space straight y right parenthesis equals straight x. log space straight x plus log left parenthesis log space straight x right parenthesis Differentiating space straight w. straight r. straight t space straight x comma fraction numerator 1 over denominator log space straight y end fraction.1 over straight y. dy over dx equals straight x.1 over straight x plus log space straight x.1 plus fraction numerator 1 over denominator log space straight x end fraction.1 over straight x therefore space dy over dx equals straight y space log space straight y open square brackets 1 plus log space straight x plus fraction numerator 1 over denominator straight x space log space straight x end fraction close square brackets equals straight x to the power of straight x to the power of straight x end exponent. straight x to the power of straight x. log space straight x open square brackets 1 plus log space straight x plus fraction numerator 1 over denominator straight x space log space straight x end fraction close square brackets

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246. Differentiate (log x)^{log x}, x > 1 w.r.t.x.

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247. If x^y = e^{x - y}, prove that

dy over dx equals fraction numerator log space straight x over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction

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248.

If space straight y equals straight x to the power of straight x to the power of straight x........ infinity end exponent end exponent space prove space that space straight x dy over dx equals fraction numerator straight y squared over denominator 1 minus straight y space log space straight x end fraction

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249.

If space straight y equals straight e to the power of straight x plus straight e to the power of straight x plus straight e to the power of straight x plus........ infinity end exponent end exponent comma space show space that space dy over dx equals fraction numerator straight y over denominator 1 minus straight y end fraction.

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250.

If space straight y equals straight a to the power of straight a to the power of straight a to the power of straight a........... infinity end exponent end exponent end exponent comma prove space that space dy over dx equals fraction numerator straight y squared log space straight y over denominator straight x left parenthesis 1 minus straight y space log space straight x. log space straight y right parenthesis end fraction

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