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Pertanyaan

f ( x ) = x 2 ( 3 x 2 − x 2 ​ ) 3 ​ ; tentukan f ( x ) .

; tentukan .

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D. Kamilia

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

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diperoleh .

diperoleh f apostrophe open parentheses x close parentheses equals open parentheses 15 x cubed minus 3 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root.

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Pembahasan

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

Turunan perkalian 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 times v end cell rightwards arrow cell f apostrophe open parentheses x close parentheses equals u apostrophe times v plus u times v apostrophe 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 f open parentheses x close parentheses equals x squared square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root, akan ditentukan f apostrophe open parentheses x close parentheses.

Misalkan:
u equals x squared rightwards arrow u apostrophe equals 2 x
begin mathsize 12px style table attributes columnalign right center left columnspacing 2px end attributes row cell v equals square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root end cell rightwards arrow cell v equals open parentheses 3 x squared minus 2 x to the power of negative 1 end exponent close parentheses to the power of 3 over 2 end exponent end cell row blank rightwards arrow cell v apostrophe equals 3 over 2 times open parentheses 3 x squared minus 2 x to the power of negative 1 end exponent close parentheses to the power of 1 half end exponent times open parentheses 6 x plus 2 x to the power of negative 2 end exponent close parentheses end cell row cell v equals square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root end cell rightwards arrow cell v apostrophe equals open parentheses 9 x plus 1 over x squared close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell end table end style

Sehingga turunannya dapat dihitung sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 2px end attributes row blank blank cell f open parentheses x close parentheses equals u times v rightwards arrow f apostrophe open parentheses x close parentheses equals u apostrophe times v plus u times v apostrophe end cell row cell f open parentheses x close parentheses end cell equals cell x squared square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell 2 x times square root of open parentheses 3 x squared minus 2 over x close parentheses cubed end root plus x squared times open parentheses 9 x plus 1 over x squared close parentheses times square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row blank equals cell 2 x open parentheses 3 x squared minus 2 over x close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root plus open parentheses 9 x cubed plus 1 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row blank equals cell open parentheses 6 x cubed minus 4 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root plus open parentheses 9 x cubed plus 1 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row blank equals cell open parentheses 6 x cubed minus 4 plus 9 x cubed plus 1 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell row cell f apostrophe open parentheses x close parentheses end cell equals cell open parentheses 15 x cubed minus 3 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root end cell end table end style

Jadi, diperoleh f apostrophe open parentheses x close parentheses equals open parentheses 15 x cubed minus 3 close parentheses square root of open parentheses 3 x squared minus 2 over x close parentheses end root.

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