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

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

  1. 0

  2. e

  3. begin mathsize 14px style straight e to the power of negative 1 end exponent end style  

  4. begin mathsize 14px style straight e to the power of 2 straight e minus 1 end exponent end style  

  5. begin mathsize 14px style straight e to the power of negative 2 straight e minus 1 end exponent end style  

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S. Intan

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

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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 begin mathsize 14px style straight u left parenthesis straight x right parenthesis equals straight e to the power of straight x end style dan begin mathsize 14px style straight v left parenthesis straight x right parenthesis equals straight x to the power of straight e end style, maka didapat begin mathsize 14px style straight u to the power of apostrophe left parenthesis straight x right parenthesis equals straight e to the power of straight x end style dan begin mathsize 14px style straight v to the power of apostrophe left parenthesis straight x right parenthesis equals straight e times straight x to the power of straight e minus 1 end exponent end style.
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 

begin mathsize 14px style straight f to the power of apostrophe left parenthesis straight e right parenthesis equals straight e to the power of straight e times straight e to the power of negative 1 minus straight e end exponent times left parenthesis straight e minus straight e right parenthesis straight f to the power of apostrophe left parenthesis straight e right parenthesis equals straight e to the power of straight e times straight e to the power of negative 1 minus straight e end exponent times 0 straight f to the power of apostrophe left parenthesis straight e right parenthesis equals 0 end style   

Jadi, jawaban yang tepat adalah A. 

 

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