RoboguruRoboguru
SD

Diketahui f(x)=(x+3) cos2 (2x−1) dan g(x)=sin2 (x−5) cos (x+1). Jika f (x) dan g (x) adalah turunan dari f(x) dan g(x), Tentukan:  a.   b. g′(x)

Pertanyaan

Diketahui f left parenthesis x right parenthesis equals open parentheses x plus 3 close parentheses space cos squared space open parentheses 2 x minus 1 close parentheses dan g left parenthesis x right parenthesis equals sin squared space open parentheses x minus 5 close parentheses space cos space open parentheses x plus 1 close parentheses. Jika f apostrophe left parenthesis x right parenthesis dan g apostrophe left parenthesis x right parenthesis adalah turunan dari f left parenthesis x right parenthesis dan g left parenthesis x right parenthesis, Tentukan: 

a. f apostrophe left parenthesis x right parenthesis 

b. g apostrophe left parenthesis x right parenthesis 

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

turunan dari fungsi f left parenthesis x right parenthesis dan g left parenthesis x right parenthesis berturut-turut adalahf apostrophe left parenthesis x right parenthesis equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos squared left parenthesis 2 x minus 1 right parenthesis minus 2 left parenthesis x plus 3 right parenthesis space sin left parenthesis 4 x minus 2 right parenthesis end cell end table dang apostrophe left parenthesis x right parenthesis equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin left parenthesis 2 x minus 10 right parenthesis space cos left parenthesis x plus 1 right parenthesis minus sin squared left parenthesis x minus 5 right parenthesis space sin left parenthesis x plus 1 right parenthesis end cell end table

Pembahasan

Untuk menjawab soal bagian (a) dan (b) gunakan aturan perkalian pada turunan yaitu 

table attributes columnalign right center left columnspacing 0px end attributes 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 end table 

Pada bagian (a) diberikan soal: 

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell left parenthesis x plus 3 right parenthesis space cos squared left parenthesis 2 x minus 1 right parenthesis end cell row blank blank blank row u equals cell x plus 3 end cell row cell u apostrophe end cell equals 1 row blank blank blank row v equals cell cos squared left parenthesis 2 x minus 1 right parenthesis end cell row cell v apostrophe end cell equals cell fraction numerator d a over denominator straight d b end fraction times fraction numerator straight d b over denominator straight d c end fraction times fraction numerator straight d c over denominator straight d x end fraction space left parenthesis aturan space rantai right parenthesis end cell row blank blank blank row blank blank cell dengan space end cell row c equals cell 2 x minus 1 end cell row cell fraction numerator straight d c over denominator straight d x end fraction end cell equals 2 row b equals cell cos space c end cell row cell fraction numerator straight d b over denominator straight d c end fraction end cell equals cell negative sin space c end cell row a equals cell b squared end cell row cell fraction numerator straight d a over denominator straight d b end fraction end cell equals cell 2 b end cell row blank blank blank row blank blank cell maka space end cell row cell v apostrophe end cell equals cell fraction numerator straight d a over denominator d b end fraction times fraction numerator straight d b over denominator straight d c end fraction times fraction numerator straight d c over denominator straight d x end fraction end cell row blank equals cell 2 b times left parenthesis negative sin space c right parenthesis times 2 end cell row blank equals cell negative 4 space cos left parenthesis 2 x minus 1 right parenthesis space sin left parenthesis 2 x minus 1 right parenthesis end cell end table   

Maka turunan dari fungsi f left parenthesis x right parenthesis 

 begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes 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 1 times cos squared left parenthesis 2 x minus 1 right parenthesis plus left parenthesis x plus 3 right parenthesis times open parentheses negative 4 close parentheses space cos left parenthesis 2 x minus 1 right parenthesis sin left parenthesis 2 x minus 1 right parenthesis end cell row blank equals cell cos squared left parenthesis 2 x minus 1 right parenthesis minus 4 left parenthesis x plus 3 right parenthesis space cos left parenthesis 2 x minus 1 right parenthesis sin left parenthesis 2 x minus 1 right parenthesis end cell row blank equals cell cos squared left parenthesis 2 x minus 1 right parenthesis minus 2 left parenthesis x plus 3 right parenthesis space times bold 2 bold space bold sin bold left parenthesis bold 2 bold italic x bold minus bold 1 bold right parenthesis bold cos bold left parenthesis bold 2 bold italic x bold minus bold 1 bold right parenthesis end cell row blank equals cell cos squared left parenthesis 2 x minus 1 right parenthesis minus 2 left parenthesis x plus 3 right parenthesis space bold sin bold 2 bold left parenthesis bold 2 bold italic x bold minus bold 1 bold right parenthesis end cell row blank equals cell cos squared left parenthesis 2 x minus 1 right parenthesis minus 2 left parenthesis x plus 3 right parenthesis space sin left parenthesis 4 x minus 2 right parenthesis end cell end table end style   

Persamaan yang ditebalkan menggunakan identitas trigonometri yaitu 2 space sin alpha times cos alpha equals sin space 2 alpha

Pada bagian (b) diberikan soal: 

table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell sin squared left parenthesis x minus 5 right parenthesis space c os left parenthesis x plus 1 right parenthesis end cell row blank blank blank row u equals cell sin squared left parenthesis x minus 5 right parenthesis end cell row cell u apostrophe end cell equals cell fraction numerator d a over denominator straight d b end fraction times fraction numerator straight d b over denominator dx end fraction space left parenthesis aturan space rantai right parenthesis end cell row blank blank blank row blank blank cell dengan space end cell row b equals cell sin left parenthesis x minus 5 right parenthesis end cell row cell fraction numerator straight d b over denominator dx end fraction end cell equals cell cos left parenthesis x minus 5 right parenthesis end cell row a equals cell b squared end cell row cell fraction numerator straight d a over denominator straight d b end fraction end cell equals cell 2 b end cell row blank blank blank row blank blank cell maka space end cell row cell u apostrophe end cell equals cell fraction numerator straight d a over denominator straight d b end fraction times fraction numerator straight d b over denominator straight d x end fraction end cell row blank equals cell 2 b times cos left parenthesis x minus 5 right parenthesis end cell row blank equals cell 2 space sin left parenthesis x minus 5 right parenthesis space cos left parenthesis x minus 5 right parenthesis space left parenthesis identitas space sin 2 alpha right parenthesis end cell row blank equals cell sin space 2 left parenthesis x minus 5 right parenthesis end cell row blank equals cell sin left parenthesis 2 x minus 10 right parenthesis space end cell row blank blank blank row v equals cell cos left parenthesis x plus 1 right parenthesis end cell row cell v apostrophe end cell equals cell negative sin left parenthesis x plus 1 right parenthesis end cell end table    

Maka turunan dari fungsi g left parenthesis x right parenthesis 

 table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe left parenthesis x right parenthesis end cell equals cell u apostrophe v plus u v apostrophe end cell row blank equals cell sin left parenthesis 2 x minus 10 right parenthesis space cos left parenthesis x plus 1 right parenthesis plus sin squared left parenthesis x minus 5 right parenthesis times negative sin left parenthesis x plus 1 right parenthesis end cell row blank equals cell sin left parenthesis 2 x minus 10 right parenthesis space cos left parenthesis x plus 1 right parenthesis minus sin squared left parenthesis x minus 5 right parenthesis space sin left parenthesis x plus 1 right parenthesis end cell end table   

Dengan demikian, turunan dari fungsi f left parenthesis x right parenthesis dan g left parenthesis x right parenthesis berturut-turut adalahf apostrophe left parenthesis x right parenthesis equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos squared left parenthesis 2 x minus 1 right parenthesis minus 2 left parenthesis x plus 3 right parenthesis space sin left parenthesis 4 x minus 2 right parenthesis end cell end table dang apostrophe left parenthesis x right parenthesis equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin left parenthesis 2 x minus 10 right parenthesis space cos left parenthesis x plus 1 right parenthesis minus sin squared left parenthesis x minus 5 right parenthesis space sin left parenthesis x plus 1 right parenthesis end cell end table

37

5.0 (3 rating)

Pertanyaan serupa

Jika  dengan , q=x sin y sin z, dan r=x cos z, tentukanlah ∂z∂w​∣∣​ x=2, y=π, z=2π​​

25

0.0

Jawaban terverifikasi

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Roboguru

Coba GRATIS Aplikasi Ruangguru

Download di Google PlayDownload di AppstoreDownload di App Gallery

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ruangguru WhatsApp

081578200000

Email info@ruangguru.com

info@ruangguru.com

Contact 02140008000

02140008000

Ikuti Kami

©2022 Ruangguru. All Rights Reserved PT. Ruang Raya Indonesia