2. Jika f(x)=a sin2 x+b cos2 x​, tunjukkan bahwa f (x)=2 f(x)(a−b) sin 2x​

Pertanyaan

2. Jika f open parentheses x close parentheses equals square root of a space sin squared space x plus b space cos squared space x end root, tunjukkan bahwa f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses a minus b close parentheses space sin space 2 x over denominator 2 space f open parentheses x close parentheses end fraction  

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

terbukti bahwa f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses a minus b close parentheses space sin space 2 x over denominator 2 space f open parentheses x close parentheses end fractionspace 

Pembahasan

Turunan fungsi f open parentheses x close parentheses equals sin space g open parentheses x close parentheses yaitu f apostrophe open parentheses x close parentheses equals cos space g open parentheses x close parentheses space g apostrophe open parentheses x close parentheses, turunan fungsi f open parentheses x close parentheses equals cos space g open parentheses x close parentheses yaituf apostrophe open parentheses x close parentheses equals negative sin space g open parentheses x close parentheses space g apostrophe open parentheses x close parentheses, dan untuk f open parentheses x close parentheses equals x to the power of n maka turunannya yaitu f apostrophe open parentheses x close parentheses equals n x to the power of n minus 1 end exponent. Untuk menyelesaikan soal di atas, perhatikan penghitungan berikut:

    f open parentheses x close parentheses equals square root of a space sin squared space x plus b space cos squared space x end root f open parentheses x close parentheses equals open parentheses a space sin squared space x plus b space cos squared space x close parentheses to the power of bevelled 1 half end exponent f apostrophe open parentheses x close parentheses equals 1 half open parentheses a space sin squared space x plus b space cos squared space x close parentheses to the power of bevelled 1 half minus 1 end exponent space open parentheses 2 a space sin space x space cos space x plus 2 b space cos space x space open parentheses negative sin space x close parentheses close parentheses f apostrophe open parentheses x close parentheses equals 1 half open parentheses a space sin squared space x plus b space cos squared space x close parentheses to the power of negative bevelled 1 half end exponent space open parentheses 2 a space sin space x space cos space x minus 2 b space cos space x space sin space x close parentheses f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses 2 a space sin space x space cos space x minus 2 b space cos space x space sin space x close parentheses over denominator 2 open parentheses open parentheses a space sin squared space x plus b space cos squared space x close parentheses to the power of begin display style bevelled 1 half end style end exponent close parentheses end fraction f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses a space 2 space sin space x space cos space x minus b space 2 space cos space x space sin space x close parentheses over denominator 2 open parentheses open parentheses a space sin squared space x plus b space cos squared space x close parentheses to the power of begin display style bevelled 1 half end style end exponent close parentheses end fraction f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses a space sin space 2 x minus b space sin space 2 x close parentheses over denominator 2 open parentheses square root of a space sin squared space x plus b space cos squared space x end root close parentheses end fraction f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses a minus b close parentheses space sin space 2 x over denominator 2 space f open parentheses x close parentheses end fraction    

Jadi, terbukti bahwa f apostrophe open parentheses x close parentheses equals fraction numerator open parentheses a minus b close parentheses space sin space 2 x over denominator 2 space f open parentheses x close parentheses end fractionspace 

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