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Pertanyaan

Dengan menggunakan aturan rantai, tentukan turunan pertama setiap fungsi berikut.

c. begin mathsize 14px style G open parentheses x close parentheses equals fraction numerator square root of 3 x squared minus 3 x plus 1 end root over denominator square root of x plus square root of x end root end fraction end style 

H. Hermawan

Master Teacher

Mahasiswa/Alumni Universitas Lampung

Jawaban terverifikasi

Jawaban

turunan dari begin mathsize 14px style G open parentheses x close parentheses equals fraction numerator square root of 3 x squared minus 3 x plus 1 end root over denominator square root of x plus square root of x end root end fraction end style adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 6 x squared square root of x plus 9 x squared minus 3 x minus 2 square root of x minus 1 over denominator open parentheses 4 x square root of x plus 4 x close parentheses square root of 3 x squared minus 3 x plus 1 end root square root of x plus square root of x end root end fraction end cell end table.

Pembahasan

Fungsi f left parenthesis x right parenthesis equals u times v dapat diturunkan dengan rumus berikut.

f apostrophe left parenthesis x right parenthesis equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction

Dengan rumus di atas, didapat perhitungan sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell square root of 3 x squared minus 3 x plus 1 end root end cell row blank equals cell open parentheses 3 x squared minus 3 x plus 1 close parentheses to the power of 1 half end exponent end cell row cell u apostrophe end cell equals cell 1 half open parentheses 3 x squared minus 3 x plus 1 close parentheses to the power of negative 1 half end exponent open parentheses 6 x minus 3 close parentheses end cell row blank equals cell fraction numerator 6 x minus 3 over denominator 2 square root of 3 x squared minus 3 x plus 1 end root end fraction end cell row blank blank blank row v equals cell square root of x plus square root of x end root end cell row blank equals cell open parentheses x plus x to the power of 1 half end exponent close parentheses to the power of 1 half end exponent end cell row cell v apostrophe end cell equals cell 1 half open parentheses x plus x to the power of 1 half end exponent close parentheses to the power of negative 1 half end exponent open parentheses 1 plus 1 half x to the power of negative 1 half end exponent close parentheses end cell row blank equals cell fraction numerator 1 plus begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator 2 square root of x plus square root of x end root end fraction end cell row blank blank blank row cell G open parentheses x close parentheses end cell equals cell fraction numerator square root of 3 x squared minus 3 x plus 1 end root over denominator square root of x plus square root of x end root end fraction end cell row cell G apostrophe open parentheses x close parentheses end cell equals cell fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction end cell row blank equals cell fraction numerator open square brackets open parentheses fraction numerator 6 x minus 3 over denominator 2 square root of 3 x squared minus 3 x plus 1 end root end fraction close parentheses open parentheses square root of x plus square root of x end root close parentheses close square brackets minus open square brackets open parentheses square root of 3 x squared minus 3 x plus 1 end root close parentheses open parentheses fraction numerator 1 plus begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator 2 square root of x plus square root of x end root end fraction close parentheses close square brackets over denominator open parentheses square root of x plus square root of x end root close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator open parentheses 6 x minus 3 close parentheses open parentheses square root of x plus square root of x end root close parentheses over denominator 2 square root of 3 x squared minus 3 x plus 1 end root end fraction end style minus begin display style fraction numerator square root of 3 x squared minus 3 x plus 1 end root open parentheses 1 plus fraction numerator 1 over denominator 2 square root of x end fraction close parentheses over denominator 2 square root of x plus square root of x end root end fraction end style over denominator x plus square root of x end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator open parentheses 6 x minus 3 close parentheses open parentheses x plus square root of x close parentheses minus open parentheses 3 x squared minus 3 x plus 1 close parentheses open parentheses fraction numerator 2 square root of x plus 1 over denominator 2 square root of x end fraction close parentheses over denominator 2 square root of 3 x squared minus 3 x plus 1 end root square root of x plus square root of x end root end fraction end style over denominator x plus square root of x end fraction end cell row blank equals cell fraction numerator 6 x squared plus 6 square root of x x minus 3 x minus 3 square root of x minus begin display style fraction numerator 6 x squared square root of x plus 3 x squared minus 6 x square root of x minus 3 x plus 2 square root of x plus 1 over denominator 2 square root of x end fraction end style over denominator open parentheses 2 x plus 2 square root of x close parentheses square root of 3 x squared minus 3 x plus 1 end root square root of x plus square root of x end root end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 12 x squared square root of x plus 12 x squared minus 6 x square root of x minus 6 x minus 6 x squared square root of x minus 3 x squared plus 6 x square root of x plus 3 x minus 2 square root of x minus 1 over denominator 2 square root of x end fraction end style over denominator open parentheses 2 x plus 2 square root of x close parentheses square root of 3 x squared minus 3 x plus 1 end root square root of x plus square root of x end root end fraction end cell row blank equals cell fraction numerator 6 x squared square root of x plus 9 x squared minus 3 x minus 2 square root of x minus 1 over denominator open parentheses 4 x square root of x plus 4 x close parentheses square root of 3 x squared minus 3 x plus 1 end root square root of x plus square root of x end root end fraction end cell end table end style

Dengan demikian, turunan dari begin mathsize 14px style G open parentheses x close parentheses equals fraction numerator square root of 3 x squared minus 3 x plus 1 end root over denominator square root of x plus square root of x end root end fraction end style adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 6 x squared square root of x plus 9 x squared minus 3 x minus 2 square root of x minus 1 over denominator open parentheses 4 x square root of x plus 4 x close parentheses square root of 3 x squared minus 3 x plus 1 end root square root of x plus square root of x end root end fraction end cell end table.

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