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Jika f(x)=x−1x+2​​ dengan x=1, maka f (x)=....

Pertanyaan

Jika begin mathsize 14px style f open parentheses x close parentheses equals square root of fraction numerator x plus 2 over denominator x minus 1 end fraction end root end style dengan begin mathsize 14px style x not equal to 1 end style, maka begin mathsize 14px style f apostrophe open parentheses x close parentheses equals.... end style 

F. Nur

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 over denominator 2 square root of x plus 2 end root times square root of left parenthesis x minus 1 right parenthesis cubed end root end fraction end cell end table end style 

Pembahasan

Sebelum menentukan turunan dari fungsi tersebut, kita ubah dulu ke dalam bentuk fungsi perpangkatan.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell square root of fraction numerator x plus 2 over denominator x minus 1 end fraction end root end cell row blank equals cell open parentheses fraction numerator x plus 2 over denominator x minus 1 end fraction close parentheses to the power of 1 half end exponent end cell row blank equals cell open parentheses x plus 2 close parentheses to the power of begin display style 1 half end style end exponent over open parentheses x minus 1 close parentheses to the power of 1 half end exponent end cell end table end style 

Jika begin mathsize 14px style f open parentheses x close parentheses equals u over v end style, berlaku begin mathsize 14px style f apostrophe open parentheses x close parentheses equals fraction numerator u apostrophe times v minus v apostrophe times u over denominator v squared end fraction end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row u equals cell open parentheses x plus 2 close parentheses to the power of 1 half end exponent end cell row cell u apostrophe end cell equals cell 1 half open parentheses x plus 2 close parentheses to the power of negative 1 half end exponent times 1 equals fraction numerator 1 over denominator 2 open parentheses x plus 2 close parentheses to the power of begin display style 1 half end style end exponent end fraction end cell row blank blank blank end table end style   

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row v equals cell left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent space end cell row cell v apostrophe end cell equals cell 1 half left parenthesis x minus 1 right parenthesis to the power of negative 1 half end exponent times 1 equals fraction numerator 1 over denominator 2 left parenthesis x minus 1 right parenthesis to the power of begin display style 1 half end style end exponent end fraction end cell end table end style 

   

begin mathsize 12px 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 u apostrophe times v minus v apostrophe times u over denominator v squared end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 1 over denominator 2 open parentheses x plus 2 close parentheses to the power of begin display style 1 half end style end exponent end fraction times open parentheses x minus 1 close parentheses to the power of 1 half end exponent minus fraction numerator 1 over denominator 2 open parentheses x minus 1 close parentheses to the power of begin display style 1 half end style end exponent end fraction times open parentheses x plus 2 close parentheses to the power of 1 half end exponent end style over denominator open parentheses open parentheses x minus 1 close parentheses to the power of begin display style 1 half end style end exponent close parentheses squared end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent over denominator 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent end fraction minus fraction numerator left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent over denominator 2 left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction end style over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent times 2 left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent minus left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent over denominator 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times 2 left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction end style over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 open parentheses x minus 1 close parentheses minus 2 open parentheses x plus 2 close parentheses over denominator 4 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction end style over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x minus 2 minus 2 x minus 4 over denominator 4 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction end style over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative 6 over denominator 4 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction end style over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator begin display style negative 3 end style over denominator 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction divided by fraction numerator x minus 1 over denominator 1 end fraction end cell row blank equals cell negative fraction numerator begin display style 3 end style over denominator 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent end fraction times fraction numerator 1 over denominator x minus 1 end fraction end cell row blank equals cell negative fraction numerator begin display style 3 end style over denominator 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 1 half end exponent times open parentheses x minus 1 close parentheses end fraction end cell row blank equals cell negative fraction numerator begin display style 3 end style over denominator 2 left parenthesis x plus 2 right parenthesis to the power of 1 half end exponent times left parenthesis x minus 1 right parenthesis to the power of 3 over 2 end exponent end fraction end cell row blank equals cell negative fraction numerator 3 over denominator 2 square root of x plus 2 end root times square root of open parentheses x minus 1 close parentheses cubed end root end fraction end cell end table end style   

Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 over denominator 2 square root of x plus 2 end root times square root of left parenthesis x minus 1 right parenthesis cubed end root end fraction end cell end table end style 

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