Pertanyaan

Diketahui f ( x ) = 9 x 3 1 − 5 x − 2 1 + 2 x 4 1 dan g ( x ) = 2 − x x 2 − 3 x + 1 . Tentukan : b. turunan kedua fungsi g

Diketahui $f (x) = 9 x^{\frac{1}{3}} - 5 x^{- \frac{1}{2}} + 2 x^{\frac{1}{4}}$ dan $g (x) = \frac{x ^{2} - 3 x + 1}{2 - x}$ . Tentukan :
b. turunan kedua fungsi $g$

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Klaim

I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

turunan kedua dari fungsi di atas adalah .

turunan kedua dari fungsi di atas adalah table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe apostrophe left parenthesis x right parenthesis end cell equals cell 2 over left parenthesis x minus 2 right parenthesis cubed end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe apostrophe left parenthesis x right parenthesis end cell equals cell 2 over left parenthesis x minus 2 right parenthesis cubed end cell end table

Pembahasan

Diketahui fungsi . Misalkan dan , maka turunan pertama dari fungsi adalah Dimisalkan dan .Turunan kedua fungsi adalah Jadi, turunan kedua dari fungsi di atas adalah .

Diketahui fungsi $g left parenthesis x right parenthesis equals fraction numerator x squared minus 3 x plus 1 over denominator 2 minus x end fraction$ . Misalkan u equals x squared minus 3 x plus 1 rightwards arrow u apostrophe equals 2 x minus 3 dan v equals 2 minus x rightwards arrow v apostrophe equals negative 1 , maka turunan pertama dari fungsi adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction end cell row blank equals cell fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis 2 minus x right parenthesis minus open parentheses x squared minus 3 x plus 1 close parentheses open parentheses negative 1 close parentheses over denominator left parenthesis 2 minus x right parenthesis squared end fraction end cell row blank equals cell fraction numerator 4 x minus 2 x squared minus 6 plus 3 x plus x squared minus 3 x plus 1 over denominator left parenthesis 2 minus x right parenthesis squared end fraction end cell row blank equals cell fraction numerator negative x squared plus 4 x minus 5 over denominator left parenthesis 2 minus x right parenthesis squared end fraction end cell end table$

Dimisalkan p equals negative x squared plus 4 x minus 5 rightwards arrow p apostrophe equals negative 2 x plus 4 dan q equals left parenthesis 2 minus x right parenthesis squared rightwards arrow q apostrophe equals 2 left parenthesis 2 minus x right parenthesis left parenthesis negative 1 right parenthesis equals 2 x minus 4 . Turunan kedua fungsi adalah

$table attributes columnalign right center left columnspacing 0px end attributes row blank equals cell fraction numerator p apostrophe q minus p q apostrophe over denominator q squared end fraction end cell row blank equals cell fraction numerator left parenthesis negative 2 x plus 4 right parenthesis left parenthesis 2 minus x right parenthesis squared plus left parenthesis x squared minus 4 x plus 5 right parenthesis left parenthesis 2 x minus 4 right parenthesis over denominator open parentheses left parenthesis 2 minus x right parenthesis squared close parentheses squared end fraction end cell row blank equals cell fraction numerator left parenthesis 2 x minus 4 right parenthesis open square brackets negative 4 plus 4 x minus x squared plus x squared minus 4 x plus 5 close square brackets over denominator left parenthesis 2 minus x right parenthesis to the power of 4 end fraction end cell row blank equals cell fraction numerator negative 2 up diagonal strike left parenthesis 2 minus x right parenthesis end strike left parenthesis 1 right parenthesis over denominator left parenthesis 2 minus x right parenthesis to the power of up diagonal strike 4 space 3 end exponent end fraction end cell row blank equals cell fraction numerator negative 2 over denominator left parenthesis 2 minus x right parenthesis cubed end fraction end cell row blank equals cell 2 over left parenthesis x minus 2 right parenthesis cubed end cell end table$

Jadi, turunan kedua dari fungsi di atas adalah table attributes columnalign right center left columnspacing 0px end attributes row cell g apostrophe apostrophe left parenthesis x right parenthesis end cell equals cell 2 over left parenthesis x minus 2 right parenthesis cubed end cell end table .