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Diketahui fungsi dan f'(x) merupakan turunan pertama dari f(x). Hasil dari f'(x) = .....

Pertanyaan

Diketahui fungsi f open parentheses x close parentheses equals fraction numerator 2 x minus 5 over denominator 3 x plus 1 end fraction comma space x not equal to negative 1 third dan f'(x) merupakan turunan pertama dari f(x). Hasil dari f'(x) = .....

  1. fraction numerator 12 x plus 17 over denominator open parentheses 3 x plus 1 close parentheses squared end fraction comma x not equal to negative 1 third

  2. 17 over open parentheses 3 x plus 1 close parentheses squared comma x not equal to negative 1 third

  3. fraction numerator negative 13 over denominator open parentheses 3 x plus 1 close parentheses squared end fraction comma x not equal to negative 1 third

  4. fraction numerator 12 x minus 13 over denominator open parentheses 3 x plus 1 close parentheses squared end fraction comma x not equal to negative 1 third

  5. fraction numerator 12 x plus 13 over denominator open parentheses 3 x plus 1 close parentheses squared end fraction comma x not equal to negative 1 third

Pembahasan Soal:

I n g a t space r u m u s  f open parentheses x close parentheses equals u over v  M a k a  f to the power of apostrophe open parentheses x close parentheses equals fraction numerator u to the power of apostrophe v minus u v apostrophe over denominator v squared end fraction  J a d i  f open parentheses x close parentheses equals fraction numerator 2 x minus 5 over denominator 3 x plus 1 end fraction  f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 2 open parentheses 3 x plus 1 close parentheses minus open parentheses 2 x minus 5 close parentheses 3 over denominator open parentheses 3 x plus 1 close parentheses squared end fraction  f to the power of apostrophe open parentheses x close parentheses equals fraction numerator 6 x plus 2 minus 6 x plus 15 over denominator open parentheses 3 x plus 1 close parentheses squared end fraction  f to the power of apostrophe open parentheses x close parentheses equals 17 over open parentheses 3 x plus 1 close parentheses squared

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

Y. Endah

Mahasiswa/Alumni Institut Teknologi Bandung

Terakhir diupdate 15 Desember 2020

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Pertanyaan yang serupa

Jika f'( x) merupakan turunan pertama dari f(x) dan diketahui f(x)=5−2xx−3​ maka f'(-1) adalah ...

Pembahasan Soal:

f left parenthesis x right parenthesis equals fraction numerator x minus 3 over denominator 5 minus 2 x end fraction

Misal :

U= x- 3

U'=1

V = 5- 2x

V=-2

  • y equals U over V rightwards arrow y apostrophe equals fraction numerator U apostrophe V minus U V apostrophe over denominator V squared end fraction  f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 left parenthesis 5 minus space 2 x right parenthesis minus left parenthesis negative 2 right parenthesis left parenthesis x minus space 3 right parenthesis over denominator left parenthesis 5 space minus space 2 x space right parenthesis squared end fraction  equals fraction numerator 5 minus space 2 x space plus 2 x minus 6 over denominator left parenthesis 5 space minus space 2 x right parenthesis squared end fraction equals fraction numerator negative 1 over denominator left parenthesis 5 space minus space 2 x right parenthesis squared end fraction  f apostrophe left parenthesis negative 1 right parenthesis equals fraction numerator negative 1 over denominator left parenthesis 5 minus space 2 left parenthesis negative 1 right parenthesis right parenthesis squared end fraction equals fraction numerator negative 1 over denominator 7 squared end fraction equals fraction numerator negative 1 over denominator 49 end fraction

 

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Roboguru

Jika y adalah turunan pertama dari y=2x2−1x2+2​ maka  adalah ...

Pembahasan Soal:

Diketahui fungsi begin mathsize 14px style y equals fraction numerator x squared plus 2 over denominator 2 x squared minus 1 end fraction end style. Misalkan,

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 x squared plus 2 space space rightwards arrow f apostrophe left parenthesis x right parenthesis equals 2 x end cell row cell g left parenthesis x right parenthesis end cell equals cell 2 x squared minus 1 rightwards arrow g apostrophe left parenthesis x right parenthesis equals 4 x end cell end table end style .

Turunan pertama dari fungsi begin mathsize 14px style y end style adalah,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction end cell row cell y apostrophe end cell equals cell fraction numerator f apostrophe left parenthesis x right parenthesis times g left parenthesis x right parenthesis minus f left parenthesis x right parenthesis times g apostrophe left parenthesis x right parenthesis over denominator g squared left parenthesis x right parenthesis end fraction end cell row cell y apostrophe end cell equals cell fraction numerator left parenthesis 2 x right parenthesis open parentheses 2 x squared minus 1 close parentheses minus open parentheses x squared plus 2 close parentheses open parentheses 4 x close parentheses over denominator open parentheses 2 x squared minus 1 close parentheses squared end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 4 x cubed minus 2 x minus 4 x cubed minus 8 x over denominator open parentheses 2 x squared minus 1 close parentheses squared end fraction end cell row cell y apostrophe end cell equals cell negative fraction numerator 10 x over denominator open parentheses 2 x squared minus 1 close parentheses squared end fraction end cell end table end style 

Jadi, table attributes columnalign right center left columnspacing 0px end attributes row cell y apostrophe end cell equals cell negative fraction numerator 10 x over denominator open parentheses 2 x squared minus 1 close parentheses squared end fraction end cell end table.

Oleh karena itu, jawaban yang tepat adalah B.

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Roboguru

Turunan pertama dari   f(x)=−x+12x+3​,x=1 adalah f'(x), maka nilai f'(2)=...

Pembahasan Soal:

f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction rightwards arrow space f apostrophe left parenthesis x right parenthesis equals fraction numerator a d minus b c over denominator left parenthesis c x plus d right parenthesis squared end fraction  f left parenthesis x right parenthesis equals fraction numerator 2 x plus 3 over denominator negative x plus 1 end fraction  M a k a  f apostrophe left parenthesis x right parenthesis equals fraction numerator 2 left parenthesis 1 right parenthesis minus left parenthesis 3 right parenthesis left parenthesis negative 1 right parenthesis over denominator left parenthesis negative x plus 1 right parenthesis squared end fraction  f apostrophe left parenthesis x right parenthesis equals fraction numerator 5 over denominator left parenthesis negative x plus 1 right parenthesis squared end fraction  f apostrophe left parenthesis 2 right parenthesis equals fraction numerator 5 over denominator left parenthesis negative 2 plus 1 right parenthesis squared end fraction  f apostrophe left parenthesis 2 right parenthesis equals 5

 

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Roboguru

Tentukan turunan pertama masing-masing fungsi berikut. c. f(x)={x(x+1)(x+2)​}3

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

Turunkan masing - masing ekspresi berikut terhadap t. b.  t33t2+4t3​

Pembahasan Soal:

Ingat!

f left parenthesis x right parenthesis equals a x to the power of n f apostrophe left parenthesis x right parenthesis equals n times a x to the power of n minus 1 end exponent  f left parenthesis x right parenthesis equals u over v f apostrophe left parenthesis x right parenthesis equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator 3 t squared plus 4 t cubed over denominator t cubed end fraction end cell row blank equals cell fraction numerator t squared left parenthesis 3 plus 4 t right parenthesis over denominator t cubed end fraction end cell row blank equals cell fraction numerator 4 t plus 3 over denominator t end fraction end cell end table

Misal:

table attributes columnalign right center left columnspacing 0px end attributes row u equals cell 4 t plus 3 end cell row cell u apostrophe end cell equals 4 row blank blank blank row v equals t row cell v apostrophe end cell equals 1 end table

Maka, 

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 u apostrophe v minus u v apostrophe over denominator v squared end fraction end cell row blank equals cell fraction numerator 4 left parenthesis t right parenthesis minus left parenthesis 4 t plus 3 right parenthesis left parenthesis 1 right parenthesis over denominator t squared end fraction end cell row blank equals cell fraction numerator 4 t minus 4 t plus 3 over denominator t squared end fraction end cell row blank equals cell 3 over t squared end cell end table

Dengan demikian, turunan dari undefined adalah t squared.

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