Pertanyaan

g ( x ) = 5 ( 2 x − 1 ) 4 x 2 ; tentukan g ( x ) .

$g (x) = \frac{x ^{2}}{5 ( 2 x - 1 ) ^{4}}$ ; tentukan $g (x)$ .

D. Kamilia

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

diperoleh .

diperoleh $g apostrophe open parentheses x close parentheses equals fraction numerator 12 x squared minus 10 x over denominator open parentheses 10 x minus 5 close parentheses fifth root of open parentheses 2 x minus 1 close parentheses to the power of 4 end root end fraction$ .

Pembahasan

Turunan pembagian dua fungsi dan aturan rantai. Diketahui , akan ditentukan . Misalkan: Sehingga turunannya dapat dihitung sebagai berikut. Jadi, diperoleh .

Turunan pembagian dua fungsi dan aturan rantai.

$table attributes columnalign right center left columnspacing 2px end attributes row cell f open parentheses x close parentheses equals u over v end cell rightwards arrow cell f apostrophe open parentheses x close parentheses equals fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction end cell row cell f open parentheses x close parentheses equals u to the power of n end cell rightwards arrow cell f apostrophe open parentheses x close parentheses equals n times u to the power of n minus 1 end exponent times u apostrophe end cell end table$

Diketahui $g open parentheses x close parentheses equals fraction numerator x squared over denominator fifth root of open parentheses 2 x minus 1 close parentheses to the power of 4 end root end fraction$ , akan ditentukan g apostrophe open parentheses x close parentheses .

Misalkan:
u equals x squared rightwards arrow u apostrophe equals 2 x
$table attributes columnalign right center left columnspacing 2px end attributes row cell v equals fifth root of left parenthesis 2 x minus 1 right parenthesis to the power of 4 end root end cell rightwards arrow cell v equals open parentheses 2 x minus 1 close parentheses to the power of 4 over 5 end exponent end cell row blank rightwards arrow cell v apostrophe equals 4 over 5 times open parentheses 2 x minus 1 close parentheses to the power of negative 1 fifth end exponent times open parentheses 2 close parentheses end cell row blank rightwards arrow cell v apostrophe equals 8 over 5 times fraction numerator 1 over denominator fifth root of open parentheses 2 x minus 1 close parentheses end root end fraction end cell row cell v equals fifth root of left parenthesis 2 x minus 1 right parenthesis to the power of 4 end root end cell rightwards arrow cell v apostrophe equals fraction numerator 8 over denominator 5 fifth root of open parentheses 2 x minus 1 close parentheses end root end fraction end cell end table$

Sehingga turunannya dapat dihitung sebagai berikut.

$table attributes columnalign right center left columnspacing 2px end attributes row blank blank cell g open parentheses x close parentheses equals u over v rightwards arrow g apostrophe open parentheses x close parentheses equals fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator x squared over denominator fifth root of open parentheses 2 x minus 1 close parentheses to the power of 4 end root end fraction end cell row cell g apostrophe open parentheses x close parentheses end cell equals cell fraction numerator 2 x times fifth root of blank minus x squared times begin display style fraction numerator 8 over denominator 5 fifth root of open parentheses 2 x minus 1 close parentheses end root end fraction end style over denominator open parentheses fifth root of blank close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x fifth root of blank times 5 fifth root of left parenthesis 2 x minus 1 right parenthesis end root minus 8 x squared over denominator 5 fifth root of left parenthesis 2 x minus 1 right parenthesis end root end fraction end style over denominator fifth root of blank end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 10 x fifth root of open parentheses 2 x minus 1 close parentheses to the power of 5 end root minus 8 x squared over denominator 5 fifth root of left parenthesis 2 x minus 1 right parenthesis end root end fraction end style over denominator fifth root of open parentheses 2 x minus 1 close parentheses to the power of 8 end root end fraction end cell row blank equals cell fraction numerator 10 x open parentheses 2 x minus 1 close parentheses minus 8 x squared over denominator 5 fifth root of left parenthesis 2 x minus 1 right parenthesis end root end fraction cross times fraction numerator 1 over denominator fifth root of open parentheses 2 x minus 1 close parentheses to the power of 8 end root end fraction end cell row blank equals cell fraction numerator 20 x squared minus 10 x minus 8 x squared over denominator 5 fifth root of open parentheses 2 x minus 1 close parentheses to the power of 9 end root end fraction end cell row blank equals cell fraction numerator 12 x squared minus 10 x over denominator 5 open parentheses 2 x minus 1 close parentheses fifth root of open parentheses 2 x minus 1 close parentheses to the power of 4 end root end fraction end cell row cell g apostrophe open parentheses x close parentheses end cell equals cell fraction numerator 12 x squared minus 10 x over denominator open parentheses 10 x minus 5 close parentheses fifth root of open parentheses 2 x minus 1 close parentheses to the power of 4 end root end fraction end cell end table$

Jadi, diperoleh $g apostrophe open parentheses x close parentheses equals fraction numerator 12 x squared minus 10 x over denominator open parentheses 10 x minus 5 close parentheses fifth root of open parentheses 2 x minus 1 close parentheses to the power of 4 end root end fraction$ .