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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

86 Views

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

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

straight x to the power of straight y equals straight e to the power of straight x minus straight y end exponent end exponent 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 rightwards double arrow space log space straight x to the power of straight y equals log space straight e to the power of straight x minus straight y end exponent therefore space straight y space log space straight x equals straight x minus straight y space space space space space space space space space space space space space space space space space space space space rightwards double arrow space straight y plus straight y space log space straight x equals straight x therefore space straight y left parenthesis 1 plus log space straight x right parenthesis equals straight x space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight y equals fraction numerator straight x over denominator 1 plus log space straight x end fraction Differentiate space both space sides space straight w. straight r. straight t. straight x comma dy over dx equals fraction numerator left parenthesis 1 plus log space straight x right parenthesis.1 minus straight x. begin display style 1 over straight x end style over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction equals fraction numerator 1 plus log space straight x minus 1 over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction 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.

82 Views

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

92 Views

Previous Year Papers

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Test Yourself

Short Answer Type

247. If xy = ex - y, prove that

247. If x^y = e^{x - y}, prove that $dy over dx equals fraction numerator log space straight x over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction$