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Jika  merupakan turunan pertama dari  maka  adalah ...

Pertanyaan

Jika fraction numerator straight d y over denominator straight d x end fraction merupakan turunan pertama dari y equals left parenthesis 2 x plus 1 right parenthesis to the power of 5 space left parenthesis x plus 1 right parenthesis maka fraction numerator straight d y over denominator straight d x end fraction adalah ... 

Pembahasan Video:

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. 

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 17 September 2021

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Pertanyaan yang serupa

Turunan pertama dari  adalah

Pembahasan Soal:

Misalkanbegin mathsize 14px style u left parenthesis x right parenthesis equals open parentheses 4 x squared plus 3 close parentheses to the power of 5 end style dan begin mathsize 14px style v left parenthesis x right parenthesis equals open parentheses 2 x minus 3 close parentheses to the power of 4 end style. Dengan menggunakan aturan rantai, turunan dari kedua fungsi tersebut adalah sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell u apostrophe left parenthesis x right parenthesis end cell equals cell 5 open parentheses 8 x close parentheses open parentheses 4 x squared plus 3 close parentheses to the power of 5 minus 1 end exponent end cell row blank equals cell 40 x open parentheses 4 x squared plus 3 close parentheses to the power of 4 end cell row cell v apostrophe left parenthesis x right parenthesis end cell equals cell 4 open parentheses 2 close parentheses open parentheses 2 x minus 3 close parentheses to the power of 4 minus 1 end exponent end cell row blank equals cell 8 open parentheses 2 x minus 3 close parentheses cubed end cell end table end style 

Turunan dari begin mathsize 14px style f left parenthesis x right parenthesis end style dapat dicari dengan menggunakan konsep turunan pada perkalian dua fungsi begin mathsize 14px style u left parenthesis x right parenthesis end style dan begin mathsize 14px style v left parenthesis x right parenthesis end style sebagai berikut:

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 fraction numerator d over denominator d x end fraction open parentheses u left parenthesis x right parenthesis times v left parenthesis x right parenthesis close parentheses end cell row blank equals cell u apostrophe left parenthesis x right parenthesis times v left parenthesis x right parenthesis plus u left parenthesis x right parenthesis times v apostrophe left parenthesis x right parenthesis end cell row blank equals cell open parentheses 40 x open parentheses 4 x squared plus 3 close parentheses to the power of 4 close parentheses open parentheses 2 x minus 3 close parentheses to the power of 4 plus open parentheses 4 x squared plus 3 close parentheses to the power of 5 open parentheses 8 open parentheses 2 x minus 3 close parentheses cubed close parentheses end cell row blank equals cell 40 x open parentheses 4 x squared plus 3 close parentheses to the power of 4 open parentheses 2 x minus 3 close parentheses to the power of 4 plus 8 open parentheses 4 x squared plus 3 close parentheses to the power of 5 open parentheses 2 x minus 3 close parentheses cubed end cell row blank equals cell 8 open parentheses 4 x squared plus 3 close parentheses to the power of 4 open parentheses 2 x minus 3 close parentheses cubed open parentheses 5 x open parentheses 2 x minus 3 close parentheses plus open parentheses 4 x squared plus 3 close parentheses close parentheses end cell row blank equals cell 8 open parentheses 4 x squared plus 3 close parentheses to the power of 4 open parentheses 2 x minus 3 close parentheses cubed open parentheses 10 x squared minus 15 x plus 4 x squared plus 3 close parentheses end cell row blank equals cell 8 open parentheses 4 x squared plus 3 close parentheses to the power of 4 open parentheses 2 x minus 3 close parentheses cubed open parentheses 14 x squared minus 15 x plus 3 close parentheses end cell end table end style 

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

 

0

Roboguru

Tentukan  jika diketahui fungsi  berikut.

Pembahasan Soal:

begin mathsize 14px style u equals open parentheses 2 x plus 5 close parentheses to the power of 4 space space space space space space space space space space space space space space space space space v equals open parentheses negative 3 x plus 1 close parentheses cubed u apostrophe equals 4 open parentheses 2 x plus 5 close parentheses cubed space space space space space space space space space space space space space space v apostrophe equals 3 open parentheses negative 3 x plus 1 close parentheses squared open parentheses negative 3 close parentheses equals negative 9 open parentheses negative 3 x plus 1 close parentheses squared f apostrophe open parentheses x close parentheses equals u apostrophe v plus v apostrophe u space space space space space space space equals 4 open parentheses 2 x plus 5 close parentheses cubed open parentheses negative 3 x plus 1 close parentheses cubed plus negative 9 open parentheses negative 3 x plus 1 close parentheses squared open parentheses 2 x plus 5 close parentheses to the power of 4 space space space space space space space equals open parentheses 2 x plus 5 close parentheses cubed open parentheses negative 3 x plus 1 close parentheses squared open parentheses 4 open parentheses negative 3 x plus 1 close parentheses minus 9 open parentheses 2 x plus 5 close parentheses close parentheses space space space space space space space equals open parentheses 2 x plus 5 close parentheses cubed open parentheses negative 3 x plus 1 close parentheses squared open parentheses negative 12 x plus 4 minus 18 x minus 45 close parentheses space space space space space space space equals open parentheses negative 30 x minus 41 close parentheses space open parentheses negative 3 x plus 1 close parentheses squared open parentheses 2 x plus 5 close parentheses cubed end style 

2

Roboguru

Tentukan turunan fungsi-fungsi berikut! b.

Pembahasan Soal:

Diketahui:

Perkalian fungsi begin mathsize 14px style f open parentheses x close parentheses equals open parentheses 2 x plus 1 close parentheses cubed open parentheses x squared minus 5 x plus 1 close parentheses squared end style.

Rumus turunan fungsi perkalian adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell u times v end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell u apostrophe times v plus u times v apostrophe end cell end table end style

Tentukan terlebih unsur-unsur yang diperlukan.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell open parentheses 2 x plus 1 close parentheses cubed end cell row cell u apostrophe end cell equals cell 3 open parentheses 2 x plus 1 close parentheses squared times 2 end cell row cell u apostrophe end cell equals cell 6 open parentheses 2 x plus 1 close parentheses squared end cell row blank blank blank row v equals cell open parentheses x squared minus 5 x plus 1 close parentheses squared end cell row cell v apostrophe end cell equals cell 2 open parentheses x squared minus 5 x plus 1 close parentheses times 2 x minus 5 end cell row cell v apostrophe end cell equals cell open parentheses 4 x minus 10 close parentheses open parentheses x squared minus 5 x plus 1 close parentheses end cell end table end style

Sehingga diperoleh:

begin mathsize 10px style 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 times v plus u times v apostrophe end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell 6 open parentheses 2 x plus 1 close parentheses squared times open parentheses x squared minus 5 x plus 1 close parentheses squared plus open parentheses 2 x plus 1 close parentheses cubed times open parentheses 4 x minus 10 close parentheses times open parentheses x squared minus 5 x plus 1 close parentheses end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell 6 open parentheses 4 x squared plus 4 x plus 1 close parentheses times open parentheses x squared minus 5 x plus 1 close parentheses squared plus open parentheses 2 x plus 1 close parentheses cubed times open parentheses 4 x minus 10 close parentheses times open parentheses x squared minus 5 x plus 1 close parentheses end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell open parentheses 24 x squared plus 24 x plus 6 close parentheses times open parentheses x squared minus 5 x plus 1 close parentheses squared plus open parentheses 2 x plus 1 close parentheses cubed times open parentheses 4 x minus 10 close parentheses times open parentheses x squared minus 5 x plus 1 close parentheses end cell end table end style

Maka, turunan fungsinya adalah begin mathsize 10px style f apostrophe open parentheses x close parentheses equals open parentheses 24 x squared plus 24 x plus 6 close parentheses open parentheses x squared minus 5 x plus 1 close parentheses squared plus open parentheses 2 x plus 1 close parentheses cubed times open parentheses 4 x minus 10 close parentheses open parentheses x squared minus 5 x plus 1 close parentheses end style.

0

Roboguru

; tentukan .

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.

0

Roboguru

Tentukan turunan pertama masing-masing fungsi berikut. c.

Pembahasan Soal:

Fungsi y=vu dapat diturunkan dengan rumus y=v2uvuv. Dengan rumus tersebut, didapat perhitungan sebagai berikut.

f(x)uuvvf(x)================{x(x+1)(x+2)}3x3(x+1)3(x+2)3(x+1)3(x+2)33(x+1)2(x+2)3+(x+1)33(x+2)23(x+2)(x+1)2(x+2)2+3(x+1)(x+1)2(x+2)2(3x+6+3x+3)(x+1)2(x+2)2(6x+9)(x+1)2(x+2)2x33x2(x3)2((6x+9)(x+1)2(x+2)2)x3(x+1)3(x+2)3(3x2)x6(6x4+9x3)(x+1)2(x+2)2(x+1)(x+2)(3x2)(x+1)2(x+2)2x6(6x4+9x3)(x+1)2(x+2)2(3x4+9x3+6x2)(x+1)2(x+2)2x6((6x4+9x3)(3x4+9x3+6x2))(x+1)2(x+2)2x6(3x46x2)(x+1)2(x+2)2x63x2(x22)(x+1)2(x+2)2x43(x22)(x+1)2(x+2)2

Dengan demikian, turunan pertama dari fungsi begin mathsize 14px style f open parentheses x close parentheses equals open curly brackets fraction numerator open parentheses x plus 1 close parentheses open parentheses x plus 2 close parentheses over denominator x end fraction close curly brackets cubed end style adalah x43(x22)(x+1)2(x+2)2.

0

Roboguru

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