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Diketahui fungsi f dinyatakan oleh f(x)=tan3x. Turunan kedua dari f yaitu f''. Tentukan : a. bentuk f(x)  b. nilai f(6π​)

Pertanyaan

Diketahui fungsi f dinyatakan oleh f left parenthesis x right parenthesis equals tan cubed space x. Turunan kedua dari f yaitu f''. Tentukan :

a. bentuk f apostrophe apostrophe left parenthesis x right parenthesis 

b. nilai f apostrophe apostrophe open parentheses straight pi over 6 close parentheses  

Pembahasan:

 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 table attributes columnalign right center left columnspacing 0px end attributes row cell v apostrophe end cell equals cell 2 times sec space x times sec space x times tan space x end cell row blank equals cell 2 times sec squared space x times tan space x end cell end table.

Sehingga turunan keduanya :

table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe apostrophe left parenthesis x right parenthesis end cell equals cell 3 open parentheses u apostrophe v plus v apostrophe u close parentheses end cell row blank equals cell 3 times 2 times tan space x times sec squared space x times sec squared space x end cell row blank blank cell plus 3 times 2 times sec squared space x times tan space x times tan squared space x end cell row blank 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   

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.

Jawaban terverifikasi

Dijawab oleh:

N. Sari

Mahasiswa/Alumni Universitas Nasional

Terakhir diupdate 07 Oktober 2021

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