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Tentukan turunan pertama fungsi-fungsi berikut!

begin mathsize 14px style f open parentheses x close parentheses equals open parentheses 5 x minus 2 x squared close parentheses open parentheses 3 x minus 1 close parentheses cubed end style   

S. Amamah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Misalkan:

begin mathsize 14px style table row cell straight u equals 5 x minus 2 x squared end cell row cell straight v equals open parentheses 3 x minus 1 close parentheses cubed end cell end table end style 

gunakan konsep turunan dan aturan rantai: 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table row cell f open parentheses x close parentheses equals a x to the power of n end cell rightwards arrow cell f apostrophe open parentheses x close parentheses equals a times n x to the power of n minus 1 end exponent end cell row cell f left parenthesis x right parenthesis equals left square bracket f open parentheses x close parentheses right square bracket to the power of n end cell rightwards arrow cell f left parenthesis x right parenthesis equals n times left square bracket f open parentheses x close parentheses right square bracket to the power of n minus 1 end exponent times f apostrophe open parentheses x close parentheses end cell end table end cell row blank blank blank end table end style

Sehingga diperoleh:

begin mathsize 14px style table row cell straight u equals 5 x minus 2 x squared end cell rightwards arrow cell straight u apostrophe equals 5 minus 4 x end cell row cell straight v equals open parentheses 3 x minus 1 close parentheses cubed end cell rightwards arrow cell straight v apostrophe equals 3 open parentheses 3 x minus 1 close parentheses squared times 3 equals 9 open parentheses 3 x minus 1 close parentheses squared end cell end table end style 

Berdasarkan konsep turunan perkalian dua fungsi maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe open parentheses x close parentheses end cell equals cell u apostrophe times v plus u times v apostrophe end cell row blank equals cell open parentheses 5 minus 4 x close parentheses open parentheses 3 x minus 1 close parentheses cubed plus open parentheses 5 x minus 2 x squared close parentheses 9 open parentheses 3 x minus 1 close parentheses squared end cell row blank equals cell open parentheses 5 minus 4 x close parentheses open parentheses 3 x minus 1 close parentheses cubed plus open parentheses 45 x minus 18 x squared close parentheses open parentheses 3 x minus 1 close parentheses squared end cell row blank equals cell open parentheses 3 x minus 1 close parentheses squared open square brackets open parentheses 5 minus 4 x close parentheses open parentheses 3 x minus 1 close parentheses plus open parentheses 45 x minus 18 x squared close parentheses close square brackets end cell row blank equals cell open parentheses 3 x minus 1 close parentheses squared open square brackets open parentheses 15 x minus 5 minus 12 x squared plus 4 x close parentheses plus open parentheses 45 x minus 18 x squared close parentheses close square brackets end cell row blank equals cell open parentheses 3 x minus 1 close parentheses squared open parentheses negative 30 x squared plus 64 x minus 5 close parentheses end cell end table end style  

Dengan demikian turunan pertama dari fungsi tersebut adalah begin mathsize 14px style f apostrophe open parentheses x close parentheses equals open parentheses 3 x minus 1 close parentheses squared open parentheses negative 30 x squared plus 64 x minus 5 close parentheses end style.

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