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Dengan menggunakan konsep turunan, tentukan turunan dari fungsi-fungsi berikut. f(x)=2x3−1​

Pertanyaan

Dengan menggunakan konsep turunan, tentukan turunan dari fungsi-fungsi berikut.

 

  1. ... 

  2. ... 

Pembahasan Video:

Pembahasan Soal:

Informasi penting: begin mathsize 14px style square root of x equals x to the power of 1 half end exponent end style

Misal

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 2 x cubed minus 1 end cell row cell fraction numerator d u over denominator d x end fraction end cell equals cell 6 x squared end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f open parentheses x close parentheses equals u to the power of 1 half end exponent end cell row cell fraction numerator d y over denominator d u end fraction end cell equals cell 1 half u to the power of negative 1 half end exponent end cell row blank equals cell fraction numerator 1 over denominator 2 u to the power of 1 half end exponent end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of u end fraction end cell end table end style

maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of apostrophe open parentheses x close parentheses end cell equals cell fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of u end fraction times 6 x squared end cell row blank equals cell fraction numerator 3 x squared over denominator square root of 2 x cubed minus 1 end root end fraction end cell end table end style

Jadi turunan dari begin mathsize 14px style square root of 2 x cubed minus 1 end root end style adalah begin mathsize 14px style fraction numerator 3 x squared over denominator square root of 2 x cubed minus 1 end root end fraction end style

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Dijawab oleh:

F. Ayudhita

Terakhir diupdate 05 Oktober 2021

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Pertanyaan yang serupa

Untuk y=3−4x​, tentukan dxdy​=…

Pembahasan Soal:

Ingat aturan rantai fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator straight d y over denominator straight d u end fraction times fraction numerator straight d u over denominator straight d x end fraction, misalkan u equals 3 minus 4 x sehingga y equals blank square root of u maka

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell square root of u end cell row y equals cell u to the power of 1 half end exponent end cell row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell fraction numerator straight d y over denominator straight d u end fraction times fraction numerator straight d u over denominator straight d x end fraction end cell row blank equals cell fraction numerator straight d left parenthesis u to the power of 1 half end exponent right parenthesis over denominator straight d u end fraction times fraction numerator straight d left parenthesis 3 minus 4 x right parenthesis over denominator straight d x end fraction end cell row blank equals cell 1 half u to the power of negative 1 half end exponent times negative 4 end cell row blank equals cell negative 2 u to the power of negative 1 half end exponent end cell row blank equals cell fraction numerator negative 2 over denominator square root of u end fraction end cell row blank equals cell fraction numerator negative 2 over denominator square root of 3 minus 4 x end root end fraction end cell end table

Oleh karena itu, jawaban yang benar adalah E.

0

Roboguru

Diketahui , maka nilai ....

Pembahasan Soal:

begin mathsize 14px style straight f open parentheses straight x close parentheses equals cube root of straight x to the power of 4 plus 5 straight x squared minus 7 end root end style 

Misalkan begin mathsize 14px style straight u equals straight x to the power of 4 plus 5 straight x squared minus 7 end style, maka begin mathsize 14px style straight u straight apostrophe equals 4 straight x cubed plus 10 straight x end style.

Sehingga turunan pertama dari fungsi tersebut yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses straight x close parentheses end cell equals cell straight u to the power of 1 third end exponent straight f to the power of straight apostrophe open parentheses straight x close parentheses equals 1 third straight u to the power of negative 2 over 3 end exponent times straight u straight apostrophe end cell row cell straight f to the power of straight apostrophe open parentheses straight x close parentheses end cell equals cell open parentheses 1 third open parentheses straight x to the power of 4 plus 5 straight x squared minus 7 close parentheses to the power of negative 2 over 3 end exponent close parentheses open parentheses 4 straight x cubed plus 10 straight x close parentheses end cell row cell straight f to the power of straight apostrophe open parentheses straight x close parentheses end cell equals cell fraction numerator open parentheses 4 straight x cubed plus 10 straight x close parentheses over denominator 3 cube root of open parentheses straight x to the power of 4 plus 5 straight x squared minus 7 close parentheses squared end root end fraction end cell end table end style 

Maka nilai dari begin mathsize 14px style straight f to the power of straight apostrophe open parentheses 1 close parentheses plus straight f to the power of straight apostrophe open parentheses negative 1 close parentheses end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f to the power of straight apostrophe open parentheses 1 close parentheses plus straight f to the power of straight apostrophe open parentheses negative 1 close parentheses end cell equals cell fraction numerator open parentheses 4 open parentheses 1 close parentheses cubed plus 10 open parentheses 1 close parentheses close parentheses over denominator 3 cube root of open parentheses 1 to the power of 4 plus 5 open parentheses 1 close parentheses squared minus 7 close parentheses squared end root end fraction plus fraction numerator open parentheses 4 open parentheses negative 1 close parentheses cubed plus 10 open parentheses negative 1 close parentheses close parentheses over denominator 3 cube root of open parentheses open parentheses negative 1 close parentheses to the power of 4 plus 5 open parentheses negative 1 close parentheses squared minus 7 close parentheses squared end root end fraction end cell row blank equals cell fraction numerator 14 over denominator 3 open parentheses 1 close parentheses end fraction minus fraction numerator 14 over denominator 3 open parentheses 1 close parentheses end fraction end cell row blank equals 0 end table end style 

Jadi, jawaban yang tepat adalah C.

1

Roboguru

Turunan pertama dari fungsi f(x)=10(5x2−3)4 adalah f(x)=...

Pembahasan Soal:

Perhatikan perhitungan berikut.

Ingat, aturan rantai pada turunan:

begin mathsize 14px style f apostrophe open parentheses x close parentheses equals fraction numerator d f open parentheses u close parentheses over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end style

Diketahui begin mathsize 14px style f open parentheses x close parentheses equals 10 open parentheses 5 x squared minus 3 close parentheses to the power of 4 end style, maka:

Misal:

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 5 x squared minus 3 end cell row cell fraction numerator d u over denominator d x end fraction end cell equals cell 2 times 5 x to the power of 2 minus 1 end exponent end cell row cell fraction numerator d u over denominator d x end fraction end cell equals cell 10 x end cell end table end style

kemudian

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses u close parentheses end cell equals cell 10 u to the power of 4 end cell row cell fraction numerator d f open parentheses u close parentheses over denominator d u end fraction end cell equals cell 4 times 10 u to the power of 4 minus 1 end exponent end cell row cell fraction numerator d f open parentheses u close parentheses over denominator d u end fraction end cell equals cell 40 u cubed end cell end table end style


Sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe open parentheses x close parentheses end cell equals cell fraction numerator d f left parenthesis u right parenthesis over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row blank equals cell 40 u cubed times 10 x end cell row blank equals cell 400 x u cubed end cell row blank equals cell 400 x open parentheses 5 x squared minus 3 close parentheses cubed end cell end table end style

Jadi, turunan pertama dari fungsi begin mathsize 14px style f open parentheses x close parentheses equals 10 open parentheses 5 x squared minus 3 close parentheses to the power of 4 end style adalah begin mathsize 14px style 400 table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 5 x squared minus 3 close parentheses end cell end table cubed end style.

Dengan demikian, jawaban yang tepat adalah E.

0

Roboguru

Carilah dxdy​ dari fungsi-fungsi berikut! y=(x−3)32​(x2+x+1)

Pembahasan Soal:

Sifat turunan fungsi: Jika begin mathsize 14px style f open parentheses x close parentheses equals u open parentheses x close parentheses times v open parentheses x close parentheses end style, maka turunannya adalah

begin mathsize 14px style f apostrophe open parentheses x close parentheses equals u apostrophe open parentheses x close parentheses times v open parentheses x close parentheses plus u open parentheses x close parentheses times v apostrophe open parentheses x close parentheses end style

Turunan dari fungsi begin mathsize 14px style y equals open parentheses x minus 3 close parentheses to the power of bevelled 2 over 3 end exponent open parentheses x squared plus x plus 1 close parentheses end style, yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell 2 over 3 open parentheses x minus 3 close parentheses to the power of negative bevelled 1 third end exponent open parentheses x squared plus x plus 1 close parentheses plus open parentheses x minus 3 close parentheses to the power of bevelled 2 over 3 end exponent open parentheses 2 x plus 1 close parentheses end cell row blank equals cell fraction numerator 2 over denominator 3 cube root of open parentheses x minus 3 close parentheses end root end fraction open parentheses x squared plus x plus 1 close parentheses plus cube root of left parenthesis x minus 3 right parenthesis squared end root open parentheses 2 x plus 1 close parentheses end cell end table end style

Dengan demikian, begin mathsize 14px style fraction numerator d y over denominator d x end fraction end style dari fungsi begin mathsize 14px style y equals open parentheses x minus 3 close parentheses to the power of bevelled 2 over 3 end exponent open parentheses x squared plus x plus 1 close parentheses end style adalah begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 over denominator 3 cube root of open parentheses x minus 3 close parentheses end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x squared plus x plus 1 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cube root of left parenthesis x minus 3 right parenthesis squared end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 2 x plus 1 close parentheses end cell end table end style

0

Roboguru

Diketahui f(x)=3(2x3+3x+3)2​ dan f(x) adalah turunan pertama f(x). Nilai f(1) adalah ...

Pembahasan Soal:

Menentukan turunan dengan aturan rantai

f(x)f(x)==aunnaun1u

 

Diketahui straight f left parenthesis straight x right parenthesis equals cube root of open parentheses 2 straight x cubed plus 3 straight x plus 3 close parentheses squared end root.

Misalkan 

uuu===2x3+3x+332x31+13x11+06x2+3

Turunan pertama dari f left parenthesis x right parenthesis adalah:

f(x)f(x)f(x)f(x)f(x)f(x)f(x)=======3(2x3+3x+3)2u3232u321u32u31u33u2u332x3+3x+32(6x2+3)332x3+3x+312x2+6 

Nilai dari straight f apostrophe left parenthesis 1 right parenthesis adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell straight f apostrophe left parenthesis 1 right parenthesis end cell equals cell fraction numerator 12 left parenthesis 1 right parenthesis squared plus 6 over denominator 3 cube root of 2 left parenthesis 1 right parenthesis cubed plus 3 left parenthesis 1 right parenthesis plus 3 end root end fraction end cell row blank equals cell fraction numerator 12 plus 6 over denominator 3 cube root of 2 plus 3 plus 3 end root end fraction end cell row blank equals cell fraction numerator 18 over denominator 3 cube root of 8 end fraction end cell row blank equals cell fraction numerator 18 over denominator 3 open parentheses 2 close parentheses end fraction end cell row blank equals cell 18 over 6 end cell row blank equals 3 end table 

Jadi, nilai straight f apostrophe left parenthesis 1 right parenthesis adalah 3.

Oleh karena itu, jawaban yang benar adalah C.

2

Roboguru

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