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if f left parenthesis x right parenthesis space equals space left parenthesis 1 plus x right parenthesis left parenthesis 1 plus x to the power of 2 space end exponent right parenthesis left parenthesis 1 plus x to the power of 4 right parenthesis................... left parenthesis 1 plus x to the power of 32 right parenthesis space t h e n space f apostrophe left parenthesis x right parenthesis space a t space x equals 1 divided by 2 space i s
Asked by ppavani198 | 19 Dec, 2018, 09:14: PM
answered-by-expert Expert Answer
f(x) = (1+x)(1+x2)(1+x4)(1+x8)(1+x16)(1+x32)
 
log f(x) = log(1+x)+log(1+x2)+log(1+x4)+log(1+x8)+(1+x16)+log(1+x32)
 
By differentiation of above,
begin mathsize 12px style fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction f apostrophe left parenthesis x right parenthesis space equals space fraction numerator 1 over denominator 1 plus x end fraction plus fraction numerator 2 x over denominator 1 plus x squared end fraction plus fraction numerator 4 x cubed over denominator 1 plus x to the power of 4 end fraction plus fraction numerator 8 x to the power of 7 over denominator 1 plus x to the power of 8 end fraction plus fraction numerator 16 x to the power of 15 over denominator 1 plus x to the power of 16 end fraction plus fraction numerator 32 x to the power of 31 over denominator 1 plus x to the power of 32 end fraction f apostrophe left parenthesis x right parenthesis space space equals space f left parenthesis x right parenthesis space open curly brackets fraction numerator begin display style 1 end style over denominator begin display style 1 plus x end style end fraction plus fraction numerator begin display style 2 x end style over denominator begin display style 1 plus x squared end style end fraction plus fraction numerator begin display style 4 x cubed end style over denominator begin display style 1 plus x to the power of 4 end style end fraction plus fraction numerator begin display style 8 x to the power of 7 end style over denominator begin display style 1 plus x to the power of 8 end style end fraction plus fraction numerator begin display style 16 x to the power of 15 end style over denominator begin display style 1 plus x to the power of 16 end style end fraction plus fraction numerator begin display style 32 x to the power of 31 end style over denominator begin display style 1 plus x to the power of 32 end style end fraction close curly brackets end style
 
to get f '(x) at x=1/2, make the substitution in the above eqn.
Answered by Thiyagarajan K | 20 Dec, 2018, 05:04: PM

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