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Continuity And Differentiability

Question
CBSEENMA12034851
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Find the derivative of the function given by

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

 

Solution
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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