Find the derivative of the function given by f(x)=(1 + x)(1 + x2

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 Multiple Choice QuestionsShort 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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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
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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).

 


Here space space straight f left parenthesis straight x right parenthesis equals left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis
therefore space space logf left parenthesis straight x right parenthesis equals log left square bracket left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis right square bracket
rightwards double arrow space logf left parenthesis straight x right parenthesis equals log left parenthesis 1 plus straight x right parenthesis plus log left parenthesis 1 plus straight x squared right parenthesis plus log left parenthesis 1 plus straight x to the power of 4 right parenthesis plus log left parenthesis 1 plus straight x to the power of 8 right parenthesis
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 space fraction numerator straight f apostrophe left parenthesis straight x right parenthesis over denominator straight f left parenthesis straight x right parenthesis end fraction equals fraction numerator 1 over denominator 1 plus straight x end fraction plus fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction plus fraction numerator 4 straight x cubed over denominator 1 plus straight x to the power of 4 end fraction plus fraction numerator 8 straight x to the power of 7 over denominator 1 plus straight x to the power of 8 end fraction
therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis open square brackets fraction numerator table row cell left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis plus 2 straight x left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis end cell row cell plus 4 straight x cubed left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus 8 right parenthesis plus 8 straight x to the power of 7 left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis end cell end table over denominator left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis end fraction close square brackets
therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis equals left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis plus 2 straight x left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis plus 4 straight x cubed left parenthesis 1 plus straight x right parenthesis
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 space space space space space space space space space space space space space space space space space left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 8 right parenthesis plus 8 straight x to the power of 7 left parenthesis 1 plus straight x right parenthesis left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight x to the power of 4 right parenthesis
therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis equals left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis plus 2 left parenthesis 1 right parenthesis left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis plus 4 left parenthesis 1 right parenthesis cubed left parenthesis 1 plus 1 right parenthesis
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 space space space space space space space space space space space space space space space space space space left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis plus 8 left parenthesis 1 right parenthesis to the power of 7 left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis
space space space space space space space space space space space space space space space space equals 8 plus 16 plus 32 plus 64 equals 120
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