Pertanyaan

Jika f ( x ) = cos ec x ⋅ sec x sec x + cos ec x , maka nilai f 2019 ( x ) ( turunan ke − 2019 ) adalah ...

Jika $f (x) = \frac{sec x + cos ec x}{cos ec x \cdot sec x}$ , maka nilai $f^{2019} (x) (turunan ke - 2019)$ adalah ...

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.

Pembahasan

Ingatlah! Turunan pertama, Di dapatkan , kemudian cari turunannya Turunan pertamanya adalah . Kemudian cari turunan kedua, ketiga, keempat dan kelima. Turunan kedua, Turunan ketiga, Turunan keempat, Turunan kelima, Karena turunan pertama dan kelima sama, maka turunan kembali ke bentuk awal setiap empat kali, maka Karena ada sisa, adalah turunan ketiga. Jadi, adalah atau . Oleh karena itu, jawaban yang benar adalah E.

Ingatlah!

$c s c left parenthesis x right parenthesis equals fraction numerator 1 over denominator sin left parenthesis x right parenthesis end fraction s e c left parenthesis x right parenthesis equals fraction numerator 1 over denominator cos left parenthesis x right parenthesis end fraction$

Turunan pertama,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses fraction numerator sec open parentheses x close parentheses plus csc open parentheses x close parentheses over denominator csc open parentheses x close parentheses sec open parentheses x close parentheses end fraction close parentheses end cell equals cell fraction numerator d over denominator d x end fraction fraction numerator fraction numerator 1 over denominator sin open parentheses x close parentheses end fraction plus sec open parentheses x close parentheses over denominator fraction numerator 1 over denominator sin open parentheses x close parentheses end fraction sec open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction fraction numerator fraction numerator 1 over denominator sin open parentheses x close parentheses end fraction plus fraction numerator 1 over denominator cos open parentheses x close parentheses end fraction over denominator fraction numerator 1 over denominator sin open parentheses x close parentheses end fraction times fraction numerator 1 over denominator cos open parentheses x close parentheses end fraction end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction fraction numerator fraction numerator 1 over denominator sin open parentheses x close parentheses end fraction plus fraction numerator 1 over denominator cos open parentheses x close parentheses end fraction over denominator fraction numerator 1 over denominator sin open parentheses x close parentheses cos open parentheses x close parentheses end fraction end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction fraction numerator fraction numerator cos open parentheses x close parentheses plus sin open parentheses x close parentheses over denominator sin open parentheses x close parentheses cos open parentheses x close parentheses end fraction over denominator fraction numerator 1 over denominator sin open parentheses x close parentheses cos open parentheses x close parentheses end fraction end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction fraction numerator open parentheses cos open parentheses x close parentheses plus sin open parentheses x close parentheses close parentheses sin open parentheses x close parentheses cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses cos open parentheses x close parentheses times thin space 1 end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction fraction numerator open parentheses cos open parentheses x close parentheses plus sin open parentheses x close parentheses close parentheses sin open parentheses x close parentheses cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses cos open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction fraction numerator open parentheses cos open parentheses x close parentheses plus sin open parentheses x close parentheses close parentheses cos open parentheses x close parentheses over denominator cos open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator d over denominator d x end fraction cos open parentheses x close parentheses plus sin open parentheses x close parentheses end cell end table$

Di dapatkan $fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses plus sin open parentheses x close parentheses close parentheses$ , kemudian cari turunannya

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses plus sin open parentheses x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses plus fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses end cell row blank equals cell negative sin open parentheses x close parentheses plus cos open parentheses x close parentheses end cell row blank equals cell sin open parentheses x close parentheses plus cos open parentheses x close parentheses end cell end table$

Turunan pertamanya adalah negative sin open parentheses x close parentheses plus cos open parentheses x close parentheses . Kemudian cari turunan kedua, ketiga, keempat dan kelima.

Turunan kedua,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses negative sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses end cell equals cell negative fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses plus fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses end cell row blank equals cell cos open parentheses x close parentheses minus sin open parentheses x close parentheses end cell end table$

Turunan ketiga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses negative cos open parentheses x close parentheses minus sin open parentheses x close parentheses close parentheses end cell equals cell negative fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses minus fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses end cell row blank equals cell negative open parentheses negative sin open parentheses x close parentheses close parentheses minus cos open parentheses x close parentheses end cell row blank equals cell sin open parentheses x close parentheses minus cos open parentheses x close parentheses end cell end table$

Turunan keempat,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses minus cos open parentheses x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses minus fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses end cell row blank equals cell cos open parentheses x close parentheses minus open parentheses negative sin open parentheses x close parentheses close parentheses end cell row blank equals cell cos open parentheses x close parentheses plus sin open parentheses x close parentheses end cell end table$

Turunan kelima,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses plus sin open parentheses x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses plus fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses end cell row blank equals cell negative sin open parentheses x close parentheses plus cos open parentheses x close parentheses end cell row blank equals cell cos open parentheses x close parentheses minus sin open parentheses x close parentheses end cell end table$

Karena turunan pertama dan kelima sama, maka turunan kembali ke bentuk awal setiap empat kali, maka

2019 equals open parentheses 504 cross times 4 close parentheses plus 3

Karena ada sisa, f to the power of 2019 left parenthesis x right parenthesis left parenthesis turunan space ke minus 2019 right parenthesis adalah turunan ketiga.

Jadi, f to the power of 2019 left parenthesis x right parenthesis left parenthesis turunan space ke minus 2019 right parenthesis adalah sin open parentheses x close parentheses minus cos open parentheses x close parentheses atau negative cos space x space plus space sin space x .

Oleh karena itu, jawaban yang benar adalah E.