Gradien garis singgung kurva y=x1 di titik (1,1) adalah ….

Pertanyaan

Gradien garis singgung kurva $begin mathsize 14px style straight y equals fraction numerator 1 over denominator square root of straight x end fraction end style$ di titik (1,1) adalah ….

R. RGFLSATU

Master Teacher

Jawaban terverifikasi

Pembahasan

Diketahui:

$begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator square root of straight x end fraction end style$

Maka,

$begin mathsize 14px style straight f left parenthesis straight x plus straight h right parenthesis equals fraction numerator 1 over denominator square root of straight x plus straight h end root end fraction end style$

Sehingga diperoleh turunan kurva tersebut adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight m left parenthesis straight x right parenthesis end cell equals cell limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator 1 over denominator square root of straight x plus straight h end root end fraction minus fraction numerator 1 over denominator square root of straight x end fraction end style over denominator straight h end fraction close parentheses end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator square root of straight x minus square root of straight x plus straight h end root over denominator square root of straight x plus straight h end root times square root of straight x end fraction end style over denominator straight h end fraction close parentheses end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style square root of straight x minus square root of straight x plus straight h end root end style over denominator straight h square root of straight x open parentheses straight x plus straight h close parentheses end root end fraction close parentheses end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style square root of straight x minus square root of straight x plus straight h end root end style over denominator straight h square root of straight x open parentheses straight x plus straight h close parentheses end root end fraction times fraction numerator begin display style square root of straight x plus square root of straight x plus straight h end root end style over denominator square root of straight x plus square root of straight x plus straight h end root end fraction close parentheses end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style straight x minus open parentheses straight x plus straight h close parentheses end style over denominator straight h square root of straight x open parentheses straight x plus straight h close parentheses end root times square root of straight x plus square root of straight x plus straight h end root end fraction close parentheses end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style negative straight h end style over denominator straight h square root of straight x open parentheses straight x plus straight h close parentheses end root times open parentheses square root of straight x plus square root of straight x plus straight h end root close parentheses end fraction close parentheses end cell row blank equals cell limit as straight h rightwards arrow 0 of open parentheses fraction numerator begin display style negative 1 end style over denominator square root of straight x open parentheses straight x plus straight h close parentheses end root times open parentheses square root of straight x plus square root of straight x plus straight h end root close parentheses end fraction close parentheses end cell row blank equals cell negative fraction numerator begin display style 1 end style over denominator square root of straight x left parenthesis straight x plus 0 right parenthesis end root times left parenthesis square root of straight x plus square root of straight x plus 0 end root right parenthesis end fraction end cell row blank equals cell negative fraction numerator begin display style 1 end style over denominator square root of straight x squared end root times left parenthesis 2 square root of straight x right parenthesis end fraction end cell row blank equals cell negative fraction numerator begin display style 1 end style over denominator 2 straight x square root of straight x end fraction end cell end table end style$

Kemudian gradien garis singgung kurva di titik (1, 1) adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight m end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis straight x right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 over denominator 2 straight x square root of straight x end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight m end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis 1 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 over denominator 2 left parenthesis 1 right parenthesis square root of 1 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table end style$