Pertanyaan

Jika f ( x ) = x e e x , maka nilai dari f'(e) adalah ....

Jika $f (x) = \frac{e ^{x}}{x ^{e}}$ , maka nilai dari f'(e) adalah ....

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S. Intan

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Mahasiswa/Alumni Institut Pertanian Bogor

Jawaban terverifikasi

Pembahasan

Ingat bahwa turunan pertama dari adalah Misalkan dan , maka didapat dan . Oleh karena itu, Oleh karena itu, didapat Jadi, jawaban yang tepat adalah A.

Ingat bahwa turunan pertama dari $begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals fraction numerator straight u left parenthesis straight x right parenthesis over denominator straight v left parenthesis straight x right parenthesis end fraction end style$ adalah

$begin mathsize 14px style straight f to the power of apostrophe left parenthesis straight x right parenthesis equals fraction numerator straight u to the power of apostrophe left parenthesis straight x right parenthesis straight v left parenthesis straight x right parenthesis minus straight u left parenthesis straight x right parenthesis straight v to the power of apostrophe left parenthesis straight x right parenthesis over denominator left parenthesis straight v left parenthesis straight x right parenthesis right parenthesis squared end fraction end style$

Misalkan dan , maka didapat dan .
Oleh karena itu,

$begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals straight e to the power of straight x over straight x to the power of straight e straight f left parenthesis straight x right parenthesis equals fraction numerator straight u left parenthesis straight x right parenthesis over denominator straight v left parenthesis straight x right parenthesis end fraction straight f to the power of apostrophe left parenthesis straight x right parenthesis equals fraction numerator straight u to the power of apostrophe left parenthesis straight x right parenthesis straight v left parenthesis straight x right parenthesis minus straight u left parenthesis straight x right parenthesis straight v to the power of apostrophe left parenthesis straight x right parenthesis over denominator left parenthesis straight v left parenthesis straight x right parenthesis right parenthesis squared end fraction straight f to the power of apostrophe left parenthesis straight x right parenthesis equals fraction numerator straight e to the power of straight x times straight x to the power of straight e minus straight e to the power of straight x times straight e times straight x to the power of straight e minus 1 end exponent over denominator left parenthesis straight x to the power of straight e right parenthesis squared end fraction straight f to the power of apostrophe left parenthesis straight x right parenthesis equals fraction numerator straight e to the power of straight x times straight x to the power of straight e minus 1 end exponent times straight x minus straight e to the power of straight x times straight e times straight x to the power of straight e minus 1 end exponent over denominator left parenthesis straight x to the power of straight e right parenthesis squared end fraction straight f to the power of apostrophe left parenthesis straight x right parenthesis equals fraction numerator straight e to the power of straight x times straight x to the power of straight e minus 1 end exponent times left parenthesis straight x minus straight e right parenthesis over denominator straight x to the power of straight e times 2 end exponent end fraction straight f to the power of apostrophe left parenthesis straight x right parenthesis equals fraction numerator straight e to the power of straight x times straight x to the power of straight e minus 1 end exponent times left parenthesis straight x minus straight e right parenthesis over denominator straight x to the power of 2 straight e end exponent end fraction straight f to the power of apostrophe left parenthesis straight x right parenthesis equals straight e to the power of straight x times straight x to the power of open parentheses straight e minus 1 close parentheses minus 2 straight e end exponent times left parenthesis straight x minus straight e right parenthesis straight f to the power of apostrophe left parenthesis straight x right parenthesis equals straight e to the power of straight x times straight x to the power of negative 1 minus straight e end exponent times left parenthesis straight x minus straight e right parenthesis end style$

Oleh karena itu, didapat

Jadi, jawaban yang tepat adalah A.