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Diketahui f(x)=(4x2−3x+5)3. Turunan pertama fungsi f adalah ...

Pertanyaan

Diketahui f open parentheses x close parentheses equals open parentheses 4 x squared minus 3 x plus 5 close parentheses cubed. Turunan pertama fungsi f adalah ...

Pembahasan Video:

Pembahasan Soal:

Gunakan konsep aturan rantai pada turunan fungsi aljabar, menentukan nilai f apostrophe left parenthesis x right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell u to the power of n end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell n times u to the power of n minus 1 end exponent times u apostrophe end cell end table

Diketahui:

f left parenthesis x right parenthesis equals left parenthesis 4 x squared minus 3 x plus 5 right parenthesis cubed

misalkan 

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 4 x squared minus 3 x plus 5 end cell row cell u apostrophe end cell equals cell 8 x minus 3 end cell end table 

 

table attributes columnalign right center left columnspacing 0px end attributes row cell u apostrophe end cell equals cell 4 times 2 x to the power of 2 minus 1 end exponent minus 3 end cell row blank equals cell 8 x minus 3 end cell end table 

sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell u cubed end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell 3 u to the power of 3 minus 1 end exponent times u apostrophe end cell row blank equals cell 3 u squared times open parentheses 8 x minus 3 close parentheses end cell row blank equals cell 3 open parentheses 4 x squared minus 3 x plus 5 close parentheses squared open parentheses 8 x minus 3 close parentheses end cell row blank equals cell open parentheses 24 x minus 9 close parentheses open parentheses 4 x squared minus 3 x plus 5 close parentheses squared end cell end table

 

Dengan demikian turunan pertama fungsi f adalah f apostrophe open parentheses x close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 24 x minus 9 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 4 x squared minus 3 x plus 5 close parentheses end cell end table squared.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 07 Oktober 2021

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Pertanyaan yang serupa

Tentukan turunan pertama dari fungsi berikut! b. g(x)=(x2+4x)4

Pembahasan Soal:

Perlu diingat bahwa jika terdapat suatu fungsi y equals open parentheses f open parentheses x close parentheses close parentheses to the power of n maka turunan pertamanya adalah y apostrophe equals n times open parentheses f open parentheses x close parentheses close parentheses to the power of n minus 1 end exponent times f apostrophe open parentheses x close parentheses.

Turunan pertama dari fungsi g left parenthesis x right parenthesis equals open parentheses x squared plus 4 x close parentheses to the power of 4:

table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell open parentheses x squared plus 4 x close parentheses to the power of 4 end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell 4 open parentheses x squared plus 4 x close parentheses to the power of 4 minus 1 end exponent times f apostrophe open parentheses x squared plus 4 x close parentheses end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell 4 open parentheses x squared plus 4 x close parentheses cubed times open parentheses 2 x to the power of 2 minus 1 end exponent plus 1 times 4 x to the power of 1 minus 1 end exponent close parentheses end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell 4 open parentheses x squared plus 4 x close parentheses cubed times open parentheses 2 x plus 4 close parentheses end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell open parentheses 8 x plus 16 close parentheses open parentheses x squared plus 4 x close parentheses cubed end cell end table  

Jadi, turunan pertama dari fungsi g left parenthesis x right parenthesis equals open parentheses x squared plus 4 x close parentheses to the power of 4 adalah Error converting from MathML to accessible text..

0

Roboguru

Turunan pertama dari f(x)=(2x2−3x+1)4 adalah f(x) = ....

Pembahasan Soal:

Diketahui : f left parenthesis x right parenthesis equals left parenthesis 2 x squared minus 3 x plus 1 right parenthesis to the power of 4

Ditanya : Turunan pertama dari fungsi aljabar ?

Jawab :
Dengan mengggunakan konsep aturan rantai pada turunan fungsi aljabar.

Misalkan : straight u equals 2 straight x squared minus 3 straight x plus 1 fraction numerator d straight y over denominator d straight u end fraction equals 4 straight x minus 3

y equals straight u to the power of 4 fraction numerator d y over denominator d u end fraction equals 4 straight u cubed

Maka turunan pertamanya adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell fraction numerator d y over denominator d x end fraction end cell row blank equals cell fraction numerator d y over denominator d u end fraction cross times fraction numerator d y over denominator d x end fraction end cell row blank equals cell 4 u cubed cross times open parentheses 4 x minus 3 close parentheses end cell row blank equals cell 4 open parentheses 2 x squared minus 3 x plus 1 close parentheses cubed cross times open parentheses 4 x minus 3 close parentheses end cell row blank equals cell open parentheses 16 x minus 12 close parentheses open parentheses 2 x squared minus 3 x plus 1 close parentheses cubed end cell end table

Dengan demikian, diperoleh f apostrophe left parenthesis x right parenthesis adalah Error converting from MathML to accessible text..

0

Roboguru

Dengan menggunakan konsep turunan, tentukan turunan dari fungsi berikut. e. f(x)=(21​x2−31​x)4

Pembahasan Soal:

Ingat!

Aturan Rantai:

begin mathsize 14px style y equals open parentheses g open parentheses x close parentheses close parentheses to the power of n comma space maka space y apostrophe equals n times open parentheses g open parentheses x close parentheses close parentheses to the power of n minus 1 end exponent times g apostrophe left parenthesis x right parenthesis end style 

Dengan:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fungsi space awal end cell row cell y apostrophe end cell equals cell turunan space pertama space fungsi space y end cell row cell g open parentheses x close parentheses end cell equals cell fungsi space yang space dipangkatkan end cell row cell g apostrophe open parentheses x close parentheses end cell equals cell turunan space pertama space fungsi space g left parenthesis x right parenthesis end cell end table end style 

Diketahui begin mathsize 14px style f left parenthesis x right parenthesis equals open parentheses 1 half x squared minus 1 third x close parentheses to the power of 4 end style, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell 1 half x squared minus 1 third x end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell 1 half times 2 x to the power of 2 minus 1 end exponent minus 1 third x to the power of 1 minus 1 end exponent end cell row cell g apostrophe left parenthesis x right parenthesis end cell equals cell x minus 1 third end cell end table end style   

Sehingga turunan fungsi tersebut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell open parentheses 1 half x squared minus 1 third x close parentheses to the power of 4 end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell n open parentheses g open parentheses x close parentheses close parentheses to the power of n minus 1 end exponent times g apostrophe left parenthesis x right parenthesis end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 4 open parentheses 1 half x squared minus 1 third x close parentheses to the power of 4 minus 1 end exponent times open parentheses x minus 1 third close parentheses end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 4 open parentheses x minus 1 third close parentheses open parentheses 1 half x squared minus 1 third x close parentheses cubed end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell open parentheses 4 x minus 4 over 3 close parentheses open parentheses 1 half x squared minus 1 third x close parentheses cubed end cell end table end style    

Jadi, turunan fungsi tersebut adalah begin mathsize 14px style open parentheses 4 x minus 4 over 3 close parentheses open parentheses 1 half x squared minus 1 third x close parentheses cubed end style.

0

Roboguru

1. Tentukan turunan pertama (dxdy​) dari fungsi berikut! a. y=(3x2+x−1)21​

Pembahasan Soal:

Perhatikan penghitungan berikut!

 begin mathsize 14px style y equals 1 over open parentheses 3 x squared plus x minus 1 close parentheses squared y equals open parentheses 3 x squared plus x minus 1 close parentheses to the power of negative 2 end exponent fraction numerator d y over denominator d x end fraction equals negative 2 open parentheses 3 x squared plus x minus 1 close parentheses to the power of negative 2 minus 1 space end exponent open parentheses 2 times 3 x plus 1 minus 0 close parentheses fraction numerator d y over denominator d x end fraction equals negative 2 open parentheses 3 x squared plus x minus 1 close parentheses to the power of negative 3 space end exponent open parentheses 6 x plus 1 close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator negative 2 open parentheses 6 x plus 1 close parentheses over denominator open parentheses 3 x squared plus x minus 1 close parentheses to the power of 3 space end exponent end fraction fraction numerator d y over denominator d x end fraction equals fraction numerator negative 12 x minus 2 over denominator open parentheses 3 x squared plus x minus 1 close parentheses to the power of 3 space end exponent end fraction space end style    

Jadi, turunan pertama dari fungsi tersebut adalah begin mathsize 14px style fraction numerator d y over denominator d x end fraction equals fraction numerator negative 12 x minus 2 over denominator open parentheses 3 x squared plus x minus 1 close parentheses to the power of 3 space end exponent end fraction space end style .   

0

Roboguru

Diketahui f(x)=(2x−3)4, f(x) merupakan turunan pertama dari f(x). Nilai dari f(3)=….

Pembahasan Soal:

Ingat rumus aturan rantai, jika terdapat begin mathsize 14px style y equals open square brackets f left parenthesis x right parenthesis close square brackets to the power of n end style maka turunan pertamanya adalah begin mathsize 14px style y apostrophe equals n open square brackets f left parenthesis x right parenthesis close square brackets to the power of n minus 1 end exponent cross times f apostrophe left parenthesis x right parenthesis end style. Diketahui pada soal begin mathsize 14px style y equals left parenthesis 2 x minus 3 right parenthesis to the power of 4 end style, maka begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x minus 3 rightwards arrow f apostrophe left parenthesis x right parenthesis equals 2 end style. Turunan pertama fungsi tersebut adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell open square brackets f left parenthesis x right parenthesis close square brackets to the power of n end cell row y equals cell left parenthesis 2 x minus 3 right parenthesis to the power of 4 end cell row cell y apostrophe end cell equals cell n left square bracket f left parenthesis x right parenthesis right square bracket to the power of n minus 1 end exponent cross times f apostrophe left parenthesis x right parenthesis end cell row cell y apostrophe end cell equals cell 4 left parenthesis 2 x minus 3 right parenthesis to the power of 4 minus 1 end exponent cross times 2 end cell row blank equals cell 4 left parenthesis 2 x minus 3 right parenthesis cubed cross times 2 space end cell row blank equals cell 8 left parenthesis 2 x minus 3 right parenthesis cubed end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 8 left parenthesis 2 x minus 3 right parenthesis cubed end cell end table end style

Untuk mencari nilai undefined, substitusikan 3 pada turunan pertama fungsi tersebut, maka

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 8 left parenthesis 2 x minus 3 right parenthesis cubed end cell row cell f apostrophe left parenthesis 3 right parenthesis end cell equals cell 8 left parenthesis 2 cross times 3 minus 3 right parenthesis cubed space space end cell row blank equals cell 8 left parenthesis 6 minus 3 right parenthesis cubed end cell row blank equals cell 8 cross times 3 cubed end cell row blank equals cell 8 cross times 27 end cell row blank equals 216 end table end style

Jadi, nilai dari undefined dari fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals left parenthesis 2 x minus 3 right parenthesis to the power of 4 end style adalah begin mathsize 14px style 216 end style.

2

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