Roboguru

f(x)=x2(3x2−x2​)3​; tentukan f(x).

Pertanyaan

f open parentheses x close parentheses equals x squared square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root; tentukan f apostrophe open parentheses x close parentheses.

Pembahasan Soal:

Turunan perkalian dua fungsi dan aturan rantai.

table attributes columnalign right center left columnspacing 2px end attributes row cell f open parentheses x close parentheses equals u times v end cell rightwards arrow cell f apostrophe open parentheses x close parentheses equals u apostrophe times v plus u times v apostrophe end cell row cell f open parentheses x close parentheses equals u to the power of n end cell rightwards arrow cell f apostrophe open parentheses x close parentheses equals n times u to the power of n minus 1 end exponent times u apostrophe end cell end table

Diketahui f open parentheses x close parentheses equals x squared square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root, akan ditentukan f apostrophe open parentheses x close parentheses.

Misalkan:
u equals x squared rightwards arrow u apostrophe equals 2 x
begin mathsize 12px style table attributes columnalign right center left columnspacing 2px end attributes row cell v equals square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root end cell rightwards arrow cell v equals open parentheses 3 x squared minus 2 x to the power of negative 1 end exponent close parentheses to the power of 3 over 2 end exponent end cell row blank rightwards arrow cell v apostrophe equals 3 over 2 times open parentheses 3 x squared minus 2 x to the power of negative 1 end exponent close parentheses to the power of 1 half end exponent times open parentheses 6 x plus 2 x to the power of negative 2 end exponent close parentheses end cell row cell v equals square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root end cell rightwards arrow cell v apostrophe equals open parentheses 9 x plus 1 over x squared close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell end table end style

Sehingga turunannya dapat dihitung sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 2px end attributes row blank blank cell f open parentheses x close parentheses equals u times v rightwards arrow f apostrophe open parentheses x close parentheses equals u apostrophe times v plus u times v apostrophe end cell row cell f open parentheses x close parentheses end cell equals cell x squared square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell 2 x times square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root plus x squared times open parentheses 9 x plus 1 over x squared close parentheses times square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row blank equals cell 2 x open parentheses 3 x squared minus 2 over x close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root plus open parentheses 9 x cubed plus 1 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row blank equals cell open parentheses 6 x cubed minus 4 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root plus open parentheses 9 x cubed plus 1 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row blank equals cell open parentheses 6 x cubed minus 4 plus 9 x cubed plus 1 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell open parentheses 15 x cubed minus 3 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell end table end style

Jadi, diperoleh f apostrophe open parentheses x close parentheses equals open parentheses 15 x cubed minus 3 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Dwi

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Jika dxdy​ merupakan turunan pertama dari y=(2x+1)5(x+1) maka  adalah ...

Pembahasan Soal:

y equals left parenthesis 2 x plus 1 right parenthesis to the power of 5 space left parenthesis x plus 1 right parenthesis

misal:

udxduvdxdv====(2x+1)55(2x+1)42=10(2x+1)4dan(x+1)1  

maka turunan fungsinya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell fraction numerator straight d u over denominator straight d x end fraction times v plus v times fraction numerator dv over denominator straight d x end fraction end cell row blank equals cell 10 open parentheses 2 x plus 1 close parentheses to the power of 4 times open parentheses x plus 1 close parentheses plus open parentheses 2 x plus 1 close parentheses to the power of 5 times open parentheses 1 close parentheses end cell row blank equals cell open parentheses 10 x plus 10 close parentheses open parentheses 2 x plus 1 close parentheses to the power of 4 plus open parentheses 2 x plus 1 close parentheses to the power of 5 end cell row blank equals cell open parentheses 2 x plus 1 close parentheses to the power of 4 open parentheses 10 x plus 10 plus 2 x plus 1 close parentheses end cell row blank equals cell open parentheses 2 x plus 1 close parentheses to the power of 4 open parentheses 12 x plus 11 close parentheses end cell end table  

Jadi, turunan pertama dari fungsi tersebut adalah Error converting from MathML to accessible text. 

 

0

Roboguru

Dengan menggunakan konsep turunan tentukan turunan dari fungsi berikut: F(x)=x3(2x+1)5

Pembahasan Soal:

Diketahui begin mathsize 14px style F open parentheses x close parentheses equals x cubed open parentheses 2 x plus 1 close parentheses to the power of 5 end style, maka:

begin mathsize 14px style u equals x cubed rightwards arrow u apostrophe equals 3 x squared v equals open parentheses 2 x plus 1 close parentheses to the power of 5 rightwards arrow v apostrophe equals 5 times open parentheses 2 x plus 1 close parentheses to the power of 5 minus 1 end exponent times 2 equals 10 open parentheses 2 x plus 1 close parentheses to the power of 4 end style 

Sehingga:

begin mathsize 14px style F apostrophe open parentheses x close parentheses equals fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction F apostrophe open parentheses x close parentheses equals fraction numerator 3 x squared times open parentheses 2 x plus 1 close parentheses to the power of 5 minus x cubed times 10 open parentheses 2 x plus 1 close parentheses to the power of 4 over denominator open parentheses open parentheses 2 x plus 1 close parentheses to the power of 5 close parentheses squared end fraction F apostrophe open parentheses x close parentheses equals fraction numerator 3 x squared times open parentheses 2 x plus 1 close parentheses to the power of 5 minus x cubed times 10 open parentheses 2 x plus 1 close parentheses to the power of 4 over denominator open parentheses 2 x plus 1 close parentheses to the power of 10 end fraction end style 

Jadi, turunan dari fungsi ​​​​​​​begin mathsize 14px style F open parentheses x close parentheses equals x cubed open parentheses 2 x plus 1 close parentheses to the power of 5 end style adalah begin mathsize 14px style fraction numerator 3 x squared times open parentheses 2 x plus 1 close parentheses to the power of 5 minus x cubed times 10 open parentheses 2 x plus 1 close parentheses to the power of 4 over denominator open parentheses 2 x plus 1 close parentheses to the power of 10 end fraction end style.

0

Roboguru

Turunan pertama dari fungsi y=(2−3x)7(2x3+1)5 adalah...

Pembahasan Soal:

y equals left parenthesis 2 minus 3 x right parenthesis to the power of 7 space left parenthesis 2 x cubed plus 1 right parenthesis to the power of 5

  • Misal : U minus left parenthesis 2 space minus space 3 x right parenthesis to the power of 7  U apostrophe equals space 7 left parenthesis 2 minus 3 x right parenthesis space to the power of 6 space end exponent left parenthesis negative 3 x right parenthesis  equals negative 21 x left parenthesis 2 minus space 3 x right parenthesis to the power of 6  V equals space left parenthesis 2 x cubed space plus space 1 right parenthesis to the power of 5  V apostrophe space equals space 5 left parenthesis space 2 x to the power of 3 space end exponent plus space 1 right parenthesis to the power of 4 space left parenthesis space 6 x squared space right parenthesis  equals space 30 x squared space left parenthesis space 2 x cubed space plus space 1 right parenthesis to the power of 4
  • y space equals space U. V rightwards arrow y apostrophe space equals space U. V space plus space U V apostrophe  y apostrophe equals negative 21 x left parenthesis 2 minus 3 x right parenthesis to the power of 6 left parenthesis 2 x cubed plus 1 right parenthesis to the power of 5 plus 30 x squared left parenthesis 2 x cubed plus 1 right parenthesis to the power of 4 left parenthesis 2 minus 3 x right parenthesis to the power of 7  equals left parenthesis 2 minus space 3 x right parenthesis to the power of 6 space end exponent left parenthesis space 2 x cubed space plus 1 right parenthesis to the power of 4 left parenthesis negative 21 x left parenthesis space x cubed plus 1 right parenthesis to the power of 4 space plus 30 x squared left parenthesis 2 minus space 3 x right parenthesis right parenthesis space

1

Roboguru

Tentukan turunan pertama fungsi-fungsi berikut. c.

Pembahasan Soal:

c. Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell u left parenthesis x right parenthesis times v left parenthesis x right parenthesis end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell u apostrophe left parenthesis x right parenthesis times v left parenthesis x right parenthesis plus v apostrophe left parenthesis x right parenthesis times u left parenthesis x right parenthesis end cell end table

Misal, u open parentheses x close parentheses equals open parentheses 2 x plus 5 close parentheses cubed dan v open parentheses x close parentheses equals open parentheses negative x plus 1 close parentheses squared. Dengan aturan rantai, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell u open parentheses x close parentheses end cell equals cell open parentheses 2 x plus 5 close parentheses cubed end cell row cell u apostrophe open parentheses x close parentheses end cell equals cell 3 open parentheses 2 x plus 5 close parentheses squared times 2 end cell row blank equals cell 6 open parentheses 2 x plus 5 close parentheses squared end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell v open parentheses x close parentheses end cell equals cell open parentheses negative x plus 1 close parentheses squared end cell row cell v apostrophe open parentheses x close parentheses end cell equals cell 2 open parentheses negative x plus 1 close parentheses times open parentheses negative 1 close parentheses end cell row blank equals cell negative 2 open parentheses negative x plus 1 close parentheses end cell end table

Kemudian, dengan menggunakan aturan turunan perkalian dua fungsi di atas, diperoleh h apostrophe open parentheses x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell left parenthesis 2 x plus 5 right parenthesis cubed left parenthesis negative x plus 1 right parenthesis squared space rightwards double arrow space h open parentheses x close parentheses equals u open parentheses x close parentheses times v open parentheses x close parentheses end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell h apostrophe open parentheses x close parentheses end cell equals cell 6 left parenthesis 2 x plus 5 right parenthesis squared times left parenthesis negative x plus 1 right parenthesis squared plus open parentheses negative 2 close parentheses left parenthesis negative x plus 1 right parenthesis times left parenthesis 2 x plus 5 right parenthesis cubed space end cell row blank equals cell 6 left parenthesis 2 x plus 5 right parenthesis squared left parenthesis negative x plus 1 right parenthesis squared minus 2 left parenthesis negative x plus 1 right parenthesis left parenthesis 2 x plus 5 right parenthesis cubed space end cell row blank equals cell left parenthesis 2 x plus 5 right parenthesis squared left parenthesis negative x plus 1 right parenthesis open square brackets 6 left parenthesis negative x plus 1 right parenthesis minus 2 left parenthesis 2 x plus 5 right parenthesis close square brackets end cell row blank equals cell left parenthesis 2 x plus 5 right parenthesis squared left parenthesis negative x plus 1 right parenthesis left parenthesis negative 6 x plus 6 minus 4 x minus 10 right parenthesis end cell row blank equals cell left parenthesis 2 x plus 5 right parenthesis squared left parenthesis negative x plus 1 right parenthesis left parenthesis negative 10 x minus 4 right parenthesis end cell end table 


Dengan demikian, turunan pertama dari fungsi h open parentheses x close parentheses di atas adalah left parenthesis 2 x plus 5 right parenthesis squared left parenthesis negative x plus 1 right parenthesis left parenthesis negative 10 x minus 4 right parenthesis.

0

Roboguru

Untuk soal nomor1-8, tentukan turunan pertama dari setiap fungsi berikut. 8. f(x)=(2−3x)4(x2−2)6​

Pembahasan Soal:

Rumus turunan untuk perkalian dua fungsi adalah sebagai berikut

f open parentheses x close parentheses equals U V rightwards arrow f apostrophe open parentheses x close parentheses equals U apostrophe V plus U V apostrophe

Rumus turunan fungsi dengan aturan rantai adalah sebagai berikut

f open parentheses x close parentheses equals a U to the power of n rightwards arrow f apostrophe open parentheses x close parentheses equals a times n times U to the power of n minus 1 end exponent times U apostrophe

Dengan menggunakan kedua konsep di atas maka

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell open parentheses 2 minus 3 x close parentheses to the power of 4 square root of open parentheses x squared minus 2 close parentheses to the power of 6 end root end cell row cell f open parentheses x close parentheses end cell equals cell open parentheses 2 minus 3 x close parentheses to the power of 4 open parentheses x squared minus 2 close parentheses to the power of 6 over 2 end exponent end cell row cell f open parentheses x close parentheses end cell equals cell open parentheses 2 minus 3 x close parentheses to the power of 4 open parentheses x squared minus 2 close parentheses cubed end cell row U equals cell open parentheses 2 minus 3 x close parentheses to the power of 4 rightwards arrow U apostrophe equals 4 open parentheses 2 minus 3 x close parentheses to the power of 4 minus 1 end exponent times open parentheses negative 3 close parentheses equals negative 12 open parentheses 2 minus 3 x close parentheses cubed end cell row V equals cell open parentheses x squared minus 2 close parentheses cubed rightwards arrow V apostrophe equals 3 times open parentheses x squared minus 2 close parentheses to the power of 3 minus 1 end exponent times open parentheses 2 x close parentheses equals 6 x open parentheses x squared minus 2 close parentheses squared end cell end table

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe open parentheses x close parentheses end cell equals cell U apostrophe V plus U V apostrophe end cell row blank equals cell negative 12 open parentheses 2 minus 3 x close parentheses cubed times open parentheses x squared minus 2 close parentheses cubed plus open parentheses 2 minus 3 x close parentheses to the power of 4 times open parentheses 6 x open parentheses x squared minus 2 close parentheses squared close parentheses end cell row blank equals cell open parentheses 2 minus 3 x close parentheses cubed open parentheses x squared minus 2 close parentheses squared open parentheses negative 12 open parentheses x squared minus 2 close parentheses plus 6 x open parentheses 2 minus 3 x close parentheses close parentheses end cell row blank equals cell open parentheses 2 minus 3 x close parentheses cubed open parentheses x squared minus 2 close parentheses squared open parentheses negative 12 x squared plus 24 plus 12 x minus 18 x squared close parentheses end cell row blank equals cell open parentheses 2 minus 3 x close parentheses cubed open parentheses x squared minus 2 close parentheses squared open parentheses negative 30 x squared plus 12 x plus 24 close parentheses end cell row blank equals cell negative 6 open parentheses 2 minus 3 x close parentheses cubed open parentheses x squared minus 2 close parentheses squared open parentheses 5 x squared minus 2 x minus 4 close parentheses end cell end table

Dengan demikian Error converting from MathML to accessible text..

0

Roboguru

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