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Given-that-f-x-x-3-1-2-1-x-2-1-x-By-using-logarithmatic-differentiation-find-the-value-of-f-1-




Question Number 147612 by ZiYangLee last updated on 22/Jul/21
Given that f(x)=(((x^3 +1)^2 (√(1+x^2 )))/(1+(√x))).  By using logarithmatic differentiation,  find the value of f ′(1).
Giventhatf(x)=(x3+1)21+x21+x.Byusinglogarithmaticdifferentiation,findthevalueoff(1).
Answered by mathmax by abdo last updated on 22/Jul/21
another way  we have  ln(f(x))=2ln(x^3 +1)+(1/2)ln(1+x^2 )−ln(1+(√x)) ⇒  ⇒((f^′ (x))/(f(x)))=((6x^2 )/(x^3  +1)) +(x/(1+x^2 ))−(1/(2(√x)(1+(√x)))) ⇒  ((f^′ (1))/(f(1)))=(6/2)+(1/2)−(1/4)=3+(1/4)=((13)/4) ⇒f^′ (1)=((13)/4)f(1)  ⇒f^′ (1)=((13)/4)×((4(√2))/2)=((13(√2))/2)
anotherwaywehaveln(f(x))=2ln(x3+1)+12ln(1+x2)ln(1+x)f(x)f(x)=6x2x3+1+x1+x212x(1+x)f(1)f(1)=62+1214=3+14=134f(1)=134f(1)f(1)=134×422=1322
Answered by mathmax by abdo last updated on 22/Jul/21
f(x)=(((x^3 +1)^2 (√(1+x^2 )))/(1+(√x))) ⇒  f^′ (x)=(({6x^2 (x^3 +1)(√(1+x^2 )) +(x/( (√(1+x^2 ))))(x^3 +1)^2 }(1+(√x))−((1/(2(√x))))(x^3 +1)^2 (√(1+x^2 )))/((1+(√x))^2 ))  ⇒f^′ (1)=(((12(√2)+(1/( (√2)))×4)(2)−(1/2)(4)(√2))/4)  =((2(14(√2))−2(√2))/4)=((26(√2))/4)=((13(√2))/2)
f(x)=(x3+1)21+x21+xf(x)={6x2(x3+1)1+x2+x1+x2(x3+1)2}(1+x)(12x)(x3+1)21+x2(1+x)2f(1)=(122+12×4)(2)12(4)24=2(142)224=2624=1322

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