Pertanyaan

Diketahui f ( x ) = x x + x x 3 x . Jika f ( x ) adalah turunan pertama dari f ( x ) , nilai f ( 1 ) adalah ...

Diketahui $f (x) = \frac{x 3 x}{x x + x}$ . Jika $f (x)$ adalah turunan pertama dari $f (x)$ , nilai $f (1)$ adalah ...

A. Armanda

Master Teacher

Mahasiswa/Alumni Universitas Indraprasta PGRI

Jawaban terverifikasi

Jawaban

.

Pembahasan

Jika suatu fungsi berbentuk , maka turunannya adalah . Karena , maka: Sehingga turunanpertama dari adalah: Substitusikan pada , sehingga: Jadi, .

Jika suatu fungsi berbentuk , maka turunannya adalah $begin mathsize 14px style fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction end style$ .

Karena $begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator x cube root of x over denominator x square root of x plus square root of x end fraction end style$ , maka:

begin mathsize 14px style u equals x cube root of x u equals x to the power of 1 times x to the power of bevelled 1 third end exponent u equals x to the power of 4 over 3 end exponent u apostrophe equals 4 over 3 x to the power of 4 over 3 minus 1 end exponent u apostrophe equals 4 over 3 x to the power of 1 third end exponent u apostrophe equals 4 over 3 cube root of x end style $begin mathsize 14px style v equals x square root of x plus square root of x v equals x to the power of 1 x to the power of 1 half end exponent plus x to the power of 1 half end exponent v equals x to the power of 3 over 2 end exponent plus x to the power of 1 half end exponent v apostrophe equals 3 over 2 x to the power of 3 over 2 minus 1 end exponent plus 1 half x to the power of 1 half minus 1 end exponent v apostrophe equals 3 over 2 x to the power of 1 half end exponent plus 1 half x to the power of negative 1 half end exponent v apostrophe equals 3 over 2 square root of x plus fraction numerator 1 over denominator 2 square root of x end fraction end style$

Sehingga turunan pertama dari adalah:

$begin mathsize 14px style f apostrophe left parenthesis x right parenthesis equals fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction f apostrophe left parenthesis x right parenthesis equals fraction numerator begin display style 4 over 3 end style cube root of x open parentheses x square root of x plus square root of x close parentheses minus x cube root of x open parentheses begin display style 3 over 2 end style square root of x plus begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style close parentheses over denominator open parentheses x square root of x plus square root of x close parentheses squared end fraction f apostrophe left parenthesis x right parenthesis equals fraction numerator begin display style 4 over 3 end style open parentheses x to the power of begin display style 1 third end style plus 1 plus begin display style 1 half end style end exponent plus x to the power of begin display style 1 third end style plus begin display style 1 half end style end exponent close parentheses minus begin display style 3 over 2 end style x to the power of 1 plus begin display style 1 third end style plus begin display style 1 half end style end exponent minus begin display style 1 half end style x to the power of 1 plus begin display style 1 third end style minus begin display style 1 half end style end exponent over denominator open parentheses x square root of x plus square root of x close parentheses squared end fraction f apostrophe left parenthesis x right parenthesis equals fraction numerator begin display style 4 over 3 end style x root index 6 of x to the power of 5 end root plus begin display style 4 over 3 end style root index 6 of x to the power of 5 end root minus begin display style 3 over 2 end style x root index 6 of x to the power of 5 end root minus begin display style 1 half end style root index 6 of x to the power of 5 end root over denominator open parentheses x square root of x plus square root of x close parentheses squared end fraction f apostrophe left parenthesis x right parenthesis equals fraction numerator begin display style fraction numerator 4 over denominator 3 root index 6 of x end fraction end style plus begin display style 5 over 6 end style root index 6 of x to the power of 5 end root over denominator open parentheses x square root of x plus square root of x close parentheses squared end fraction end style$

Substitusikan pada , sehingga:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator begin display style 4 over 3 end style root index 6 of x plus begin display style 5 over 6 end style root index 6 of x to the power of 5 end root over denominator open parentheses x square root of x plus square root of x close parentheses squared end fraction end cell row cell f apostrophe left parenthesis 1 right parenthesis end cell equals cell fraction numerator begin display style fraction numerator 4 over denominator 3 root index 6 of 1 end fraction end style plus begin display style 5 over 6 end style root index 6 of 1 to the power of 5 end root over denominator open parentheses 1 square root of 1 plus square root of 1 close parentheses squared end fraction end cell row cell f apostrophe left parenthesis 1 right parenthesis end cell equals cell fraction numerator begin display style 4 over 3 end style plus begin display style 5 over 6 end style over denominator 2 squared end fraction end cell row cell f apostrophe left parenthesis 1 right parenthesis end cell equals cell fraction numerator 13 over denominator 6 times 4 end fraction end cell row cell f apostrophe left parenthesis 1 right parenthesis end cell equals cell 13 over 24 end cell end table end style$

Jadi, .