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Diketahui f(0) = 1 dan f'(0) = 2. Jika , maka g'(0...

Diketahui f(0) = 1 dan f'(0) = 2. Jika begin mathsize 12px style g open parentheses x close parentheses equals 1 over open parentheses 2 f open parentheses x close parentheses minus 1 close parentheses cubed end style, maka g'(0) = ...

  1. - 12

  2. - 6

  3. 6

  4. 8

  5. 12

Jawaban:

Perhatikan sifat aturan rantai

Definisi: Jika g(x) = y, maka g'(x) = begin mathsize 12px style g apostrophe open parentheses x close parentheses equals fraction numerator d y over denominator d a end fraction times fraction numerator d a over denominator d x end fraction end style

Dan salah satu sifat eksponensial, yaitu begin mathsize 12px style a to the power of negative 3 end exponent equals 1 over a cubed end style

Sehingga dari soal dapat diperoleh begin mathsize 12px style g open parentheses x close parentheses equals 1 over open parentheses 2 f open parentheses x close parentheses minus 1 close parentheses cubed equals open parentheses 2 f open parentheses x close parentheses minus 1 close parentheses to the power of negative 3 end exponent end style

 

Pemecahan masalah menggunakan aturan rantai

Misalkan:

a = 2 . f(x) - 1 ... (1)

begin mathsize 12px style fraction numerator d a over denominator d x end fraction equals 2 times f apostrophe open parentheses x close parentheses space horizontal ellipsis space open parentheses 2 close parentheses    y equals a cubed  fraction numerator d y over denominator d x end fraction equals negative 3 a to the power of negative 4 end exponent space horizontal ellipsis space open parentheses 3 close parentheses end style

 

Masukkan persamaan 2 dan 3 ke definisi aturan rantai, menjadi:

begin mathsize 12px style g apostrophe open parentheses x close parentheses equals negative 3 a to the power of negative 4 end exponent times 2 f apostrophe open parentheses x close parentheses  g apostrophe open parentheses x close parentheses equals negative 3 times open parentheses 2 f open parentheses x close parentheses minus 1 close parentheses to the power of negative 4 end exponent times 2 f apostrophe open parentheses x close parentheses  g apostrophe open parentheses x close parentheses equals negative 6 f apostrophe open parentheses x close parentheses open parentheses 2 f open parentheses x close parentheses minus 1 close parentheses to the power of negative 4 end exponent end style

 

Penyelesaian soal

begin mathsize 12px style g apostrophe open parentheses 0 close parentheses equals negative 6 f apostrophe open parentheses 0 close parentheses open parentheses 2. f open parentheses 0 close parentheses minus 1 close parentheses to the power of negative 4 end exponent  g apostrophe open parentheses 0 close parentheses equals negative 6 times 2 open parentheses 2 times 1 minus 1 close parentheses to the power of negative 4 end exponent  g apostrophe open parentheses 0 close parentheses equals negative 6 times 2 open parentheses 1 close parentheses to the power of negative 4 end exponent  g apostrophe open parentheses 0 close parentheses equals negative 6 times 2  g apostrophe open parentheses 0 close parentheses equals negative 12 end style

 

 

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