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Pertanyaan

Dengan menggunakan konsep turunan, tentukan turunan dari fungsi-fungsi berikut.

 

P. Anggrayni

Master Teacher

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Pembahasan

Perhatikan bahwa

merupakan bentuk perkalian dua fungsi.

Turunan dari perkalian dua fungsi dirumuskan dengan

begin mathsize 14px style f open parentheses x close parentheses equals u open parentheses x close parentheses times v open parentheses x close parentheses rightwards arrow f to the power of apostrophe open parentheses x close parentheses equals u to the power of apostrophe open parentheses x close parentheses times v open parentheses x close parentheses plus u open parentheses x close parentheses times v apostrophe left parenthesis x right parenthesis end style

Maka misalkan

u open parentheses x close parentheses equals x cubed dan v open parentheses x close parentheses equals open parentheses 2 x plus 1 close parentheses to the power of 5.

Dengan menggunakan sifat turunan 

f open parentheses x close parentheses equals a x to the power of n rightwards arrow f apostrophe open parentheses x close parentheses equals a times n times x to the power of n minus 1 end exponent

diperoleh

u apostrophe open parentheses x close parentheses equals 3 x to the power of 3 minus 1 end exponent equals 3 x squared.

Selanjutnya, untuk menentukan turunan pertama dari v open parentheses x close parentheses equals open parentheses 2 x plus 1 close parentheses to the power of 5, kita akan menggunakan aturan rantai

f open parentheses x close parentheses equals a times open parentheses p open parentheses x close parentheses close parentheses to the power of n rightwards arrow f apostrophe open parentheses x close parentheses equals a times n times open parentheses p open parentheses x close parentheses close parentheses to the power of n minus 1 end exponent times open parentheses p apostrophe open parentheses x close parentheses close parentheses.

Misalkan

p open parentheses x close parentheses equals 2 x plus 1

maka

table attributes columnalign right center left columnspacing 0px end attributes row cell p apostrophe open parentheses x close parentheses end cell equals cell 2 x to the power of 1 minus 1 end exponent plus 0 end cell row blank equals cell 2 x to the power of 0 plus 0 end cell row blank equals cell 2 times open parentheses 1 close parentheses plus 0 end cell row blank equals 2 end table

Karena

table attributes columnalign right center left columnspacing 0px end attributes row cell v open parentheses x close parentheses end cell equals cell open parentheses 2 x plus 1 close parentheses to the power of 5 end cell row cell v open parentheses x close parentheses end cell equals cell open square brackets p open parentheses x close parentheses close square brackets to the power of 5 end cell end table

maka

table attributes columnalign right center left columnspacing 0px end attributes row cell v apostrophe open parentheses x close parentheses end cell equals cell 5 times open square brackets p open parentheses x close parentheses close square brackets to the power of 5 minus 1 end exponent times p apostrophe open parentheses x close parentheses end cell row blank equals cell 5 times open parentheses 2 x plus 1 close parentheses to the power of 4 times 2 end cell row blank equals cell 10 open parentheses 2 x plus 1 close parentheses to the power of 4 end cell end table

Oleh karena itu

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 open parentheses x close parentheses v open parentheses x close parentheses plus u open parentheses x close parentheses v apostrophe open parentheses x close parentheses end cell row blank equals cell open parentheses 3 x squared close parentheses open parentheses 2 x plus 1 close parentheses to the power of 5 plus x cubed open parentheses 10 open parentheses 2 x plus 1 close parentheses to the power of 4 close parentheses end cell row blank equals cell open parentheses 2 x plus 1 close parentheses to the power of 4 left square bracket 3 x squared open parentheses 2 x plus 1 close parentheses plus 10 x cubed right square bracket end cell row blank equals cell open parentheses 2 x plus 1 close parentheses to the power of 4 left square bracket 6 x cubed plus 3 x squared plus 10 x cubed right square bracket end cell row blank equals cell open parentheses 2 x plus 1 close parentheses to the power of 4 left parenthesis 3 x squared plus 16 x cubed right parenthesis end cell end table   


Dengan demikian, turunan pertama dari fungsi begin mathsize 14px style f open parentheses x close parentheses equals x cubed open parentheses 2 x plus 1 close parentheses to the power of 5 end style adalah

f apostrophe open parentheses x close parentheses equals open parentheses 2 x plus 1 close parentheses to the power of 4 left parenthesis 3 x squared plus 16 x cubed right parenthesis

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