Pertanyaan

Tentukan turunan dari fungsi-fungsi trigonometri berikut. f. h ( x ) = sin x + cos x cos x

Tentukan turunan dari fungsi-fungsi trigonometri berikut.

f. $h (x) = \frac{cos x}{sin x + cos x}$

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G. Albiah

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Mahasiswa/Alumni Universitas Galuh Ciamis

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Pembahasan

Ingatlah! Maka, Cari turunan Cari turunan Maka, Menyederhanakan, Maka, Karena Maka, Jadi, turunan dari adalah .

Ingatlah!

$open parentheses f over g close parentheses to the power of apostrophe equals fraction numerator f thin space apostrophe times g minus g apostrophe times f over denominator g squared end fraction$

Maka,

$begin mathsize 12px style fraction numerator d over denominator d x end fraction open parentheses fraction numerator cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses plus cos open parentheses x close parentheses end fraction close parentheses equals fraction numerator fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses minus fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses cos open parentheses x close parentheses over denominator open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses squared end fraction end style$

Cari turunan $fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses$

$fraction numerator d over denominator d x end fraction open parentheses cos open parentheses x close parentheses close parentheses equals negative sin open parentheses x close parentheses$

Cari turunan $fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses$

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

Maka,

Menyederhanakan,

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator d over denominator d x end fraction open parentheses fraction numerator cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses plus cos open parentheses x close parentheses end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses negative sin open parentheses x close parentheses close parentheses open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses minus open parentheses cos open parentheses x close parentheses minus sin open parentheses x close parentheses close parentheses cos open parentheses x close parentheses over denominator open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses squared end fraction end cell row blank equals cell fraction numerator negative sin open parentheses x close parentheses cos open parentheses x close parentheses minus sin squared open parentheses x close parentheses minus open parentheses cos open parentheses x close parentheses minus sin open parentheses x close parentheses close parentheses cos open parentheses x close parentheses over denominator cos squared open parentheses x close parentheses plus 2 cos open parentheses x close parentheses sin open parentheses x close parentheses plus sin squared open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator negative sin left parenthesis x right parenthesis cos left parenthesis x right parenthesis minus sin squared left parenthesis x right parenthesis minus cos open parentheses x close parentheses cos open parentheses x close parentheses plus cos open parentheses x close parentheses sin open parentheses x close parentheses over denominator cos squared open parentheses x close parentheses plus 2 cos open parentheses x close parentheses sin open parentheses x close parentheses plus sin squared open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator up diagonal strike negative sin left parenthesis x right parenthesis cos left parenthesis x right parenthesis end strike minus sin squared left parenthesis x right parenthesis minus cos squared open parentheses x close parentheses up diagonal strike plus cos left parenthesis x right parenthesis sin left parenthesis x right parenthesis end strike over denominator cos squared left parenthesis x right parenthesis plus 2 cos left parenthesis x right parenthesis sin left parenthesis x right parenthesis plus sin squared left parenthesis x right parenthesis end fraction end cell row blank equals cell fraction numerator negative sin squared left parenthesis x right parenthesis minus cos squared left parenthesis x right parenthesis over denominator cos squared left parenthesis x right parenthesis plus 2 cos left parenthesis x right parenthesis sin left parenthesis x right parenthesis plus sin squared left parenthesis x right parenthesis end fraction end cell end table$

Karena

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative sin squared open parentheses x close parentheses minus cos squared open parentheses x close parentheses over denominator cos squared open parentheses x close parentheses plus 2 cos open parentheses x close parentheses sin open parentheses x close parentheses plus sin squared open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator negative cos squared open parentheses x close parentheses minus sin squared open parentheses x close parentheses over denominator cos squared open parentheses x close parentheses plus sin squared open parentheses x close parentheses plus 2 cos open parentheses x close parentheses sin open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator negative cos squared open parentheses x close parentheses minus sin squared open parentheses x close parentheses over denominator cos squared open parentheses x close parentheses plus sin squared open parentheses x close parentheses plus sin open parentheses 2 x close parentheses end fraction end cell row blank equals cell fraction numerator negative open parentheses 1 minus sin squared open parentheses x close parentheses close parentheses minus sin squared open parentheses x close parentheses over denominator 1 minus sin squared open parentheses x close parentheses plus sin open parentheses 2 x close parentheses plus sin squared open parentheses x close parentheses end fraction end cell row blank equals cell fraction numerator negative open parentheses 1 minus sin squared open parentheses x close parentheses close parentheses minus sin squared open parentheses x close parentheses over denominator up diagonal strike negative sin squared open parentheses x close parentheses end strike plus sin open parentheses 2 x close parentheses plus up diagonal strike sin squared open parentheses x close parentheses end strike plus 1 end fraction end cell row blank equals cell fraction numerator negative open parentheses 1 close parentheses minus open parentheses negative sin squared open parentheses x close parentheses close parentheses minus sin squared open parentheses x close parentheses over denominator sin open parentheses 2 x close parentheses plus 1 end fraction end cell row blank equals cell fraction numerator negative 1 plus up diagonal strike sin squared open parentheses x close parentheses end strike minus up diagonal strike sin squared open parentheses x close parentheses end strike over denominator sin open parentheses 2 x close parentheses plus 1 end fraction end cell row blank equals cell fraction numerator negative 1 over denominator sin open parentheses 2 x close parentheses plus 1 end fraction end cell row blank blank blank end table$

Jadi, turunan dari $h left parenthesis x right parenthesis equals fraction numerator cos space x over denominator sin space x space plus space cos space x end fraction$ adalah $h apostrophe left parenthesis x right parenthesis equals negative fraction numerator 1 over denominator sin open parentheses 2 x close parentheses plus 1 end fraction$ .