Pertanyaan

Jika f ( x ) adalah turunan kedua dari fungsi f ( x ) = sin 4 3 x , nilai f ( 6 π ) = ...

Jika $f (x)$ adalah turunan kedua dari fungsi $f (x) = sin^{4} 3 x$ , nilai $f (\frac{π}{6}) = ...$

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.

Pembahasan

Turunan pertama , Ingatlah! Maka, Turunan kedua, Ingatlah! Maka, Cari turunan Maka, Kemudian mencari turunan Maka, Di dapatkan : Cari nilainilai Jadi, nilai adalah . Oleh karena itu, jawaban yang benar adalah E .

Turunan pertama f left parenthesis x right parenthesis equals sin to the power of 4 3 x , Ingatlah!

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell u to the power of 4 end cell row u equals cell sin space 3 x end cell end table$

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses sin to the power of 4 open parentheses 3 x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d u end fraction open parentheses u to the power of 4 close parentheses fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 4 u to the power of 4 minus 1 end exponent fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 4 u cubed fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 4 open parentheses sin open parentheses 3 x close parentheses close parentheses cubed fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 4 open parentheses sin open parentheses 3 x close parentheses close parentheses cubed cos open parentheses 3 x close parentheses fraction numerator d over denominator d x end fraction open parentheses 3 x close parentheses end cell row blank equals cell 4 open parentheses sin open parentheses 3 x close parentheses close parentheses cubed cos open parentheses 3 x close parentheses 3 fraction numerator d over denominator d x end fraction open parentheses x close parentheses end cell row blank equals cell 4 open parentheses sin open parentheses 3 x close parentheses close parentheses cubed cos open parentheses 3 x close parentheses 3 times thin space 1 end cell row blank equals cell 4 open parentheses sin open parentheses 3 x close parentheses close parentheses cubed cos open parentheses 3 x close parentheses times thin space 3 end cell row blank equals cell 12 sin cubed open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses end cell end table$

Turunan kedua,

$fraction numerator d over denominator d x end fraction open parentheses 12 sin cubed open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses close parentheses equals 12 fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses close parentheses$

Ingatlah!

Maka,

$12 fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses close parentheses equals 12 open parentheses fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses close parentheses cos open parentheses 3 x close parentheses plus fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses sin cubed open parentheses 3 x close parentheses close parentheses$

Cari turunan $fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell u cubed end cell row u equals cell sin space 3 x end cell end table$

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d u end fraction open parentheses u cubed close parentheses fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 3 u to the power of 3 minus 1 end exponent fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 3 u squared fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 3 open parentheses sin open parentheses 3 x close parentheses close parentheses squared fraction numerator d over denominator d x end fraction open parentheses sin open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 3 open parentheses sin open parentheses 3 x close parentheses close parentheses squared cos open parentheses 3 x close parentheses fraction numerator d over denominator d x end fraction open parentheses 3 x close parentheses end cell row blank equals cell 3 open parentheses sin open parentheses 3 x close parentheses close parentheses squared cos open parentheses 3 x close parentheses times thin space 3 end cell row blank equals cell 3 open parentheses sin open parentheses 3 x close parentheses close parentheses squared cos open parentheses 3 x close parentheses times thin space 3 end cell row blank equals cell 9 sin squared open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses end cell end table$

Kemudian mencari turunan $fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell cos space left parenthesis u right parenthesis end cell row u equals cell space 3 x end cell end table$

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses end cell equals cell fraction numerator d over denominator d u end fraction open parentheses cos open parentheses u close parentheses close parentheses fraction numerator d over denominator d x end fraction open parentheses 3 x close parentheses end cell row blank equals cell sin open parentheses u close parentheses fraction numerator d over denominator d x end fraction open parentheses 3 x close parentheses end cell row blank equals cell negative sin open parentheses 3 x close parentheses fraction numerator d over denominator d x end fraction open parentheses 3 x close parentheses end cell row blank equals cell negative sin left parenthesis 3 x right parenthesis 3 fraction numerator d over denominator d x end fraction open parentheses x close parentheses end cell row blank equals cell negative sin open parentheses 3 x close parentheses 3 times thin space 1 end cell row blank equals cell negative sin open parentheses 3 x close parentheses times thin space 3 end cell end table$

Di dapatkan :

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 12 fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 12 open parentheses fraction numerator d over denominator d x end fraction open parentheses sin cubed open parentheses 3 x close parentheses close parentheses cos open parentheses 3 x close parentheses plus fraction numerator d over denominator d x end fraction open parentheses cos open parentheses 3 x close parentheses close parentheses sin cubed open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 12 open parentheses 9 sin squared open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses cos open parentheses 3 x close parentheses minus sin open parentheses 3 x close parentheses times thin space 3 sin cubed open parentheses 3 x close parentheses close parentheses end cell row blank equals cell 12 open parentheses 9 sin squared open parentheses 3 x close parentheses cos squared open parentheses 3 x close parentheses minus 3 sin to the power of 4 open parentheses 3 x close parentheses close parentheses end cell end table$

Cari nilai nilai f apostrophe apostrophe open parentheses straight pi over 6 close parentheses

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 12 open parentheses 9 sin squared open parentheses 3 open parentheses pi over 6 close parentheses close parentheses cos squared open parentheses 3 open parentheses pi over 6 close parentheses close parentheses minus 3 sin to the power of 4 open parentheses 3 open parentheses pi over 6 close parentheses close parentheses close parentheses end cell row blank equals cell 12 open parentheses 9 sin squared open parentheses 3 times pi over 6 close parentheses cos squared open parentheses 3 times pi over 6 close parentheses minus 3 sin to the power of 4 open parentheses 3 times pi over 6 close parentheses close parentheses end cell row blank equals cell 12 open parentheses 9 sin squared open parentheses fraction numerator 3 pi over denominator 6 end fraction close parentheses cos squared open parentheses fraction numerator 3 pi over denominator 6 end fraction close parentheses minus 3 sin to the power of 4 open parentheses fraction numerator 3 pi over denominator 6 end fraction close parentheses close parentheses end cell row blank equals cell 12 open parentheses 9 sin squared open parentheses pi over 2 close parentheses cos squared open parentheses pi over 2 close parentheses minus 3 sin to the power of 4 open parentheses straight pi over 2 close parentheses close parentheses end cell row blank equals cell 12 open parentheses 9 times 1 times 0 times negative 3 times 1 close parentheses end cell row blank equals cell 12 open parentheses 0 minus 3 close parentheses end cell row blank equals cell 12 left parenthesis negative 3 right parenthesis end cell row blank equals cell negative 36 end cell end table$

Jadi, nilai f apostrophe apostrophe open parentheses straight pi over 6 close parentheses adalah negative 36 .

Oleh karena itu, jawaban yang benar adalah E.