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Turunan pertama dari  adalah...

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Turunan pertama dari f open parentheses x close parentheses equals open parentheses 3 x squared plus 4 close parentheses to the power of 5 times open parentheses 2 x minus 1 close parentheses to the power of 4 adalah...

  1. open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 240 x close parentheses 

  2. open parentheses 3 x plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 30 x plus 8 close parentheses 

  3. open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 18 x squared minus 6 x plus 8 close parentheses 

  4. open parentheses 3 x squared plus 4 x close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 36 x squared minus 30 x minus 32 close parentheses 

  5. open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 84 x squared minus 30 x plus 32 close parentheses 

Pembahasan Soal:

Pada soal berikut termasuk turunan pada perkalian dua fungsi, dapat menggunakan rumus seperti berikut.

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell U times V end cell row y equals cell U apostrophe times V plus V apostrophe times U end cell row blank blank blank row cell f open parentheses x close parentheses end cell equals cell open parentheses 3 x squared plus 4 close parentheses to the power of 5 times open parentheses 2 x minus 1 close parentheses to the power of 4 end cell row U equals cell open parentheses 3 x squared plus 4 close parentheses to the power of 5 end cell row cell U apostrophe end cell equals cell 5 times 6 x times open parentheses 3 x squared plus 4 close parentheses to the power of 4 end cell row blank equals cell 30 x times open parentheses 3 x squared plus 4 close parentheses to the power of 4 end cell row V equals cell open parentheses 2 x minus 1 close parentheses to the power of 4 end cell row cell V apostrophe end cell equals cell 4 times 2 times open parentheses 2 x minus 1 close parentheses cubed end cell row blank equals cell 8 times open parentheses 2 x minus 1 close parentheses cubed end cell row blank blank blank row cell f open parentheses x close parentheses end cell equals cell open parentheses 3 x squared plus 4 close parentheses to the power of 5 times open parentheses 2 x minus 1 close parentheses to the power of 4 end cell row y equals cell U apostrophe times V plus V apostrophe times U end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell 30 x times open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses to the power of 4 plus 8 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 3 x squared plus 4 close parentheses to the power of 5 end cell row blank equals cell 2 open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open curly brackets 15 x open parentheses 2 x minus 1 close parentheses plus 4 open parentheses 3 x squared plus 4 close parentheses close curly brackets end cell row blank equals cell 2 open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 30 x squared minus 15 x plus 12 x squared plus 16 close parentheses end cell row blank equals cell 2 open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 42 x squared minus 15 x plus 16 close parentheses end cell row blank equals cell open parentheses 3 x squared plus 4 close parentheses to the power of 4 times open parentheses 2 x minus 1 close parentheses cubed times open parentheses 84 x squared minus 30 x plus 32 close parentheses end cell end table 

Oleh karena itu, jawaban yang tepat adalah E

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Mahasiswa/Alumni Universitas Negeri Jakarta

Terakhir diupdate 01 Mei 2021

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Pembahasan Soal:

Ingat:

begin mathsize 14px style f left parenthesis x right parenthesis equals a x to the power of n comma space maka space f apostrophe left parenthesis x right parenthesis equals a times n x to the power of n minus 1 end exponent end style 

Diketahui fungsi begin mathsize 14px style y equals 4 times left parenthesis 2 x plus 1 right parenthesis times x to the power of 4 end style , maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 4 times left parenthesis 2 x plus 1 right parenthesis times x to the power of 4 end cell row blank equals cell 8 x to the power of 5 plus 4 x to the power of 4 end cell row cell y apostrophe end cell equals cell 8 times 5 x to the power of 5 minus 1 end exponent plus 4 times 4 x to the power of 4 minus 1 end exponent end cell row blank equals cell 40 x to the power of 4 plus 16 x cubed end cell end table end style 

Jadi, turunan fungsi begin mathsize 14px style y equals 4 times left parenthesis 2 x plus 1 right parenthesis times x to the power of 4 end style adalah begin mathsize 14px style y apostrophe equals 40 x to the power of 4 plus 16 x cubed end style.

 

0

Roboguru

Turunan pertama dari adalah ...

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell space f left parenthesis x right parenthesis end cell equals cell x squared left parenthesis 3 x space – space 1 right parenthesis cubed end cell row blank blank blank row cell f left parenthesis x right parenthesis end cell equals cell u times v end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell u apostrophe v plus u v apostrophe end cell row blank blank blank row u equals cell x squared end cell row cell u apostrophe end cell equals cell 2 x end cell row blank blank blank row v equals cell left parenthesis 3 x space – space 1 right parenthesis cubed end cell row cell v apostrophe end cell equals cell 3 times 3 left parenthesis 3 x minus 1 right parenthesis to the power of 3 minus 1 end exponent end cell row cell space v apostrophe end cell equals cell 9 left parenthesis 3 x minus 1 right parenthesis squared space end cell row blank blank blank row cell f apostrophe left parenthesis x right parenthesis end cell equals cell u apostrophe v plus u v apostrophe end cell row blank equals cell 2 x left parenthesis 3 x space – space 1 right parenthesis cubed plus x squared times open parentheses 9 left parenthesis 3 x minus 1 right parenthesis squared close parentheses end cell row blank equals cell x open parentheses 3 x minus 1 close parentheses squared open parentheses 2 open parentheses 3 x minus 1 close parentheses plus 9 x close parentheses end cell row blank equals cell x left parenthesis 3 x minus 1 right parenthesis squared open parentheses 6 x minus 2 plus 9 x close parentheses space end cell row blank equals cell x left parenthesis 3 x minus 1 right parenthesis squared open parentheses 15 x minus 2 close parentheses end cell row blank blank blank end table 

Oleh karena itu, jawaban yang benar adalah B.

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Roboguru

d. Jika  dan , tentukan .

Pembahasan Soal:

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Roboguru

Carilah  dari fungsi-fungsi berikut!

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

Dari fungsi begin mathsize 14px style y equals open parentheses square root of 6 minus x cubed close parentheses open parentheses x squared plus 3 close parentheses end style, misalkan:

u left parenthesis x right parenthesis equals square root of 6 minus x cubed v left parenthesis x right parenthesis equals x squared plus 3

Dengan menggunakan rumus turunan di atas, trunan dari fungsi begin mathsize 14px style y equals open parentheses square root of 6 minus x cubed close parentheses open parentheses x squared plus 3 close parentheses end style, yaitu

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell open parentheses square root of 6 minus x cubed close parentheses open parentheses x squared plus 3 close parentheses end cell row cell y apostrophe end cell equals cell negative 3 x squared open parentheses x squared plus 3 close parentheses plus open parentheses square root of 6 minus x cubed close parentheses open parentheses 2 x close parentheses end cell row blank equals cell negative 3 x to the power of 4 minus 9 x squared plus 2 square root of 6 x minus 2 x to the power of 4 end cell row blank equals cell negative 5 x to the power of 4 minus 9 x squared plus 2 square root of 6 x end cell end table 

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 square root of 6 minus x cubed close parentheses open parentheses x squared plus 3 close parentheses end style adalah Error converting from MathML to accessible text..

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Roboguru

Jika , , , maka nilai dari  adalah ....

Pembahasan Soal:

Rumus turunan pada perkalian dua fungsi:

y equals straight U times straight V maka straight y apostrophe equals straight U apostrophe times straight V plus straight V apostrophe times straight U

Diketahui f open parentheses x close parentheses equals g open parentheses x close parentheses square root of xg open parentheses 9 close parentheses equals 4g apostrophe open parentheses 9 close parentheses equals 8, sehingga:

table attributes columnalign right center left columnspacing 2px end attributes row straight U equals cell g open parentheses x close parentheses end cell row cell straight U apostrophe end cell equals cell g apostrophe open parentheses x close parentheses end cell row straight V equals cell square root of x equals x to the power of 1 half end exponent end cell row cell straight V apostrophe end cell equals cell 1 half x to the power of negative 1 half end exponent end cell end table

Menggunakan konsep turunan pada perkalian dua fungsi dapat diperoleh:

table attributes columnalign right center left columnspacing 2px end attributes row cell f left parenthesis x right parenthesis end cell equals cell g open parentheses x close parentheses square root of x end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell g apostrophe open parentheses x close parentheses times square root of x plus 1 half x to the power of negative 1 half end exponent times g open parentheses x close parentheses end cell row cell f apostrophe open parentheses 9 close parentheses end cell equals cell g apostrophe open parentheses 9 close parentheses times square root of 9 plus 1 half 9 to the power of negative 1 half end exponent times g open parentheses 9 close parentheses end cell row cell f apostrophe open parentheses 9 close parentheses end cell equals cell 8 times 3 plus fraction numerator 1 over denominator 2 square root of 9 end fraction times 4 end cell row blank equals cell 24 plus 1 over 6 times 4 end cell row blank equals cell 24 plus 2 over 3 end cell row blank equals cell fraction numerator 72 plus 2 over denominator 3 end fraction end cell row blank equals cell 74 over 3 end cell row blank equals cell 24 2 over 3 end cell end table

Jadi, nilai dari f apostrophe open parentheses 9 close parentheses adalah 24 2 over 3.

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