Pertanyaan

Diketahui fungsi f dinyatakan oleh f ( x ) = tan 3 x . Turunan kedua dari f yaitu f''. Tentukan : a. bentuk f ( x ) b. nilai f ( 6 π )

Diketahui fungsi f dinyatakan oleh $f (x) = tan^{3} x$ . Turunan kedua dari f yaitu f''. Tentukan :

a. bentuk $f (x)$

b. nilai $f (\frac{π}{6})$

N. Sari

Master Teacher

Mahasiswa/Alumni Universitas Nasional

Jawaban terverifikasi

Pembahasan

Ingat aturan rantai turunan fungsi trigonometri : Diketahui : Misalkan , maka . Misalkan , kita nyatakan dalam . Maka . Sehingga turunan pertamanya : Diketahui : Misalkan , maka . Misalkan , maka . Sehingga turunan keduanya : Maka, . Nilai : Maka, nilai .

Ingat aturan rantai turunan fungsi trigonometri :

$fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction$

Diketahui f left parenthesis x right parenthesis equals tan cubed space x :

Misalkan u equals tan space x , maka $fraction numerator d u over denominator d x end fraction equals sec squared space x$ .

Misalkan y equals tan cubed space x , kita nyatakan dalam y equals u cubed . Maka $fraction numerator d y over denominator d u end fraction equals 3 u squared$ .

Sehingga turunan pertamanya :

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d y over denominator d x end fraction end cell equals cell fraction numerator d y over denominator d u end fraction cross times fraction numerator d u over denominator d x end fraction end cell row blank equals cell 3 u squared times sec to the power of 2 space end exponent x end cell row blank equals cell 3 times tan squared space x times sec to the power of 2 space end exponent x end cell end table$

Diketahui table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell 3 times tan squared space x times sec to the power of 2 space end exponent x end cell end table :

Misalkan table attributes columnalign right center left columnspacing 0px end attributes row u equals cell tan squared space x end cell end table , maka u apostrophe equals 2 times tan space x times sec squared space x .

Misalkan table attributes columnalign right center left columnspacing 0px end attributes row v equals cell sec to the power of 2 space end exponent x end cell end table , maka .

Sehingga turunan keduanya :

Maka, table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe apostrophe left parenthesis x right parenthesis end cell equals cell 6 times tan space x times sec to the power of 4 space x plus 6 times tan cubed space x times sec squared space x end cell end table .

Nilai f apostrophe apostrophe open parentheses straight pi over 6 close parentheses :

$table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe apostrophe left parenthesis x right parenthesis end cell equals cell 6 times tan space x times sec to the power of 4 space x plus 6 times tan cubed space x times sec squared space x end cell row cell f apostrophe apostrophe open parentheses straight pi over 6 close parentheses end cell equals cell 6 times tan space open parentheses straight pi over 6 close parentheses times sec to the power of 4 space open parentheses straight pi over 6 close parentheses end cell row blank blank cell plus 6 times tan cubed space open parentheses straight pi over 6 close parentheses times sec squared space open parentheses straight pi over 6 close parentheses end cell row blank equals cell 6 times tan space 30 degree times sec to the power of 4 space 30 degree end cell row blank blank cell plus 6 times tan cubed space 30 degree times sec squared space 30 degree end cell row blank equals cell 6 cross times fraction numerator square root of 3 over denominator 3 end fraction cross times open parentheses fraction numerator 2 over denominator square root of 3 end fraction close parentheses to the power of 4 end cell row blank blank cell plus 6 cross times open parentheses fraction numerator square root of 3 over denominator 3 end fraction close parentheses cubed cross times open parentheses fraction numerator 2 over denominator square root of 3 end fraction close parentheses squared end cell row blank equals cell 6 cross times fraction numerator square root of 3 over denominator 3 end fraction cross times 16 over 9 plus 6 cross times fraction numerator 3 square root of 3 over denominator 27 end fraction cross times 4 over 3 end cell row blank equals cell fraction numerator 32 square root of 3 over denominator 9 end fraction plus fraction numerator 8 square root of 3 over denominator 9 end fraction end cell row blank equals cell 40 over 9 square root of 3 end cell end table$

Maka, nilai table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe apostrophe open parentheses straight pi over 6 close parentheses end cell equals cell 40 over 9 square root of 3 end cell end table .