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Continuity and Differentiability

Short Answer Type

251.

If space straight x to the power of straight y equals straight y to the power of straight x comma space prove space that space dy over dx equals fraction numerator begin display style straight y over straight x end style minus log space straight y over denominator begin display style straight x over straight y end style minus log space straight x end fraction

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252.

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

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253.

Find space dy over dx when space xy equals straight e to the power of straight x minus straight y end exponent.

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254.

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255. $If space straight y equals square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis dy over dx plus straight y equals 0$

Here space If space straight y equals square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root space therefore space log space straight y equals log open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses to the power of 1 half end exponent space space space space space space space rightwards double arrow space log space straight y equals 1 half log space open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses rightwards double arrow space log space straight y equals 1 half left square bracket log left parenthesis 1 minus straight x right parenthesis minus log left parenthesis 1 plus 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 comma space space space space space 1 over straight y dy over dx equals 1 half open square brackets fraction numerator 1 over denominator 1 minus straight x end fraction left parenthesis negative 1 right parenthesis minus fraction numerator 1 over denominator 1 plus straight x end fraction close square brackets therefore space 1 over straight y dy over dx equals negative 1 half open square brackets fraction numerator 1 over denominator 1 minus straight x end fraction plus fraction numerator 1 over denominator 1 plus straight x end fraction close square brackets rightwards double arrow space 1 over straight y dy over dx equals negative 1 half open square brackets fraction numerator 1 plus straight x plus negative straight x over denominator left parenthesis 1 minus straight x right parenthesis left parenthesis 1 plus straight x right parenthesis end fraction close square brackets rightwards double arrow space 1 over straight y dy over dx equals negative 1 half open square brackets fraction numerator 2 over denominator 1 minus straight x squared end fraction close square brackets space space space space rightwards double arrow space 1 over straight y dy over dx equals negative fraction numerator 1 over denominator 1 minus straight x squared end fraction therefore left parenthesis 1 minus straight x squared right parenthesis dy over dx equals negative straight y therefore left parenthesis 1 minus straight x squared right parenthesis dy over dx plus straight y equals 0

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256.

If space straight f left parenthesis straight x right parenthesis equals open parentheses fraction numerator 3 plus straight x over denominator 1 plus straight x end fraction close parentheses to the power of 2 plus 3 straight x end exponent comma space find space straight f apostrophe left parenthesis 0 right parenthesis

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257.

Differentiate space square root of fraction numerator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x squared plus 4 right parenthesis over denominator 3 straight x squared plus 4 straight x plus 5 end fraction end root straight w. straight r. straight t. straight x

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258.

Differentiate space square root of fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 2 right parenthesis over denominator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 5 right parenthesis end fraction end root straight w. straight r. straight t. straight x

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259. Differentiate : (x + 3)² (x + 4)³ (x + 5)^4.w.r.t.x.

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260.

Find the derivative of the function given by

f(x)=(1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)and hence find f'(1).

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