Pertanyaan

Gradien garis singgung f ( x ) = x 3 − x − 3 di titik ( x 1 , y 1 ) adalah ....

Gradien garis singgung $f (x) = x^{3} - x^{- 3}$ di titik $(x_{1}, y_{1})$ adalah ....

$begin mathsize 14px style 3 x subscript 1 superscript 2 minus fraction numerator 3 over denominator x subscript 1 superscript 4 end fraction end style$
$begin mathsize 14px style 3 x subscript 1 superscript 2 plus fraction numerator 3 over denominator x subscript 1 superscript 4 end fraction end style$

S. Nur

Master Teacher

Jawaban terverifikasi

Pembahasan

Ingat bahwa gradien garis singgung fungsipada titik adalah sebagai berikut. Kemudian, diberikan permisalan sebagai berikut. A = h → 0 lim h ( ( x 1 + h ) 3 − x 1 3 ) dan B = h → 0 lim h ( ( x 1 + h ) 3 1 − x 1 3 1 ) Oleh karena itu, didapat . Kita selesaikan A dan B tersebut hingga diperoleh hasil berikut ini. dan Dengan demikian, didapat garis singgung pada titik tersebut adalah sebagai berikut. Jadi, jawaban yang tepat adalah D.

Ingat bahwa gradien garis singgung fungsi pada titik adalah sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator f left parenthesis x subscript 1 plus h right parenthesis minus f left parenthesis x subscript 1 right parenthesis over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator open parentheses left parenthesis x subscript 1 plus h right parenthesis cubed minus left parenthesis x subscript 1 plus h right parenthesis to the power of negative 3 end exponent close parentheses minus open parentheses left parenthesis x subscript 1 right parenthesis cubed minus left parenthesis x subscript 1 right parenthesis to the power of negative 3 end exponent close parentheses over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator open parentheses left parenthesis x subscript 1 plus h right parenthesis cubed minus fraction numerator 1 over denominator left parenthesis x subscript 1 plus h right parenthesis cubed end fraction close parentheses minus open parentheses x subscript 1 superscript 3 minus fraction numerator 1 over denominator x subscript 1 superscript 3 end fraction close parentheses over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator left parenthesis x subscript 1 plus h right parenthesis cubed minus begin display style fraction numerator 1 over denominator left parenthesis x subscript 1 plus h right parenthesis cubed end fraction end style minus x subscript 1 superscript 3 plus begin display style fraction numerator 1 over denominator x subscript 1 superscript 3 end fraction end style over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator open parentheses left parenthesis x subscript 1 plus h right parenthesis cubed minus x subscript 1 superscript 3 close parentheses minus begin display style open parentheses fraction numerator 1 over denominator left parenthesis x subscript 1 plus h right parenthesis cubed end fraction minus fraction numerator 1 over denominator x subscript 1 superscript 3 end fraction close parentheses end style over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator open parentheses left parenthesis x subscript 1 plus h right parenthesis cubed minus x subscript 1 superscript 3 close parentheses over denominator h end fraction minus stack text lim end text with h rightwards arrow 0 below fraction numerator begin display style open parentheses fraction numerator 1 over denominator left parenthesis x subscript 1 plus h right parenthesis cubed end fraction minus fraction numerator 1 over denominator x subscript 1 superscript 3 end fraction close parentheses end style over denominator h end fraction end cell end table end style$

Kemudian, diberikan permisalan sebagai berikut.

$A = h \to 0 lim \frac{( ( x _{1} + h ) ^{3} - x _{1}^{3} )}{h}$

dan

$B = h \to 0 lim \frac{( \frac{1}{( x _{1} + h ) ^{3}} - \frac{1}{x _{1}^{3}} )}{h}$

Oleh karena itu, didapat .

Kita selesaikan A dan B tersebut hingga diperoleh hasil berikut ini.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row A equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator left parenthesis left parenthesis x subscript 1 plus h right parenthesis cubed minus x subscript 1 superscript 3 right parenthesis over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator open parentheses x subscript 1 superscript 3 plus 3 x subscript 1 superscript 2 h plus 3 x subscript 1 h squared plus h cubed minus x subscript 1 superscript 3 close parentheses over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator left parenthesis 3 x subscript 1 superscript 2 plus 3 x subscript 1 h plus h squared right parenthesis times h over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of open parentheses 3 x subscript 1 superscript 2 plus 3 x subscript 1 h plus h squared close parentheses end cell row blank equals cell 3 x subscript 1 superscript 2 plus 3 x subscript 1 left parenthesis 0 right parenthesis plus left parenthesis 0 right parenthesis squared end cell row blank equals cell 3 x subscript 1 superscript 2 end cell end table end style$

dan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row B equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator begin display style open parentheses fraction numerator 1 over denominator left parenthesis x subscript 1 plus h right parenthesis cubed end fraction minus fraction numerator 1 over denominator x subscript 1 superscript 3 end fraction close parentheses end style over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below fraction numerator begin display style open parentheses fraction numerator x subscript 1 superscript 3 minus left parenthesis x subscript 1 plus h right parenthesis cubed right parenthesis over denominator left parenthesis x subscript 1 plus h right parenthesis cubed space x subscript 1 superscript 3 right parenthesis end fraction close parentheses end style over denominator h end fraction end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below open parentheses fraction numerator x subscript 1 superscript 3 minus left parenthesis x subscript 1 plus h right parenthesis cubed over denominator h left parenthesis x subscript 1 plus h right parenthesis cubed space x subscript 1 superscript 3 end fraction close parentheses end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below open parentheses fraction numerator x subscript 1 superscript 3 minus open parentheses x subscript 1 superscript 3 plus 3 x subscript 1 superscript 2 h plus 3 x subscript 1 h squared plus h cubed close parentheses over denominator h left parenthesis x subscript 1 plus h right parenthesis cubed space x subscript 1 superscript 3 end fraction close parentheses end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below open parentheses fraction numerator negative 3 x subscript 1 superscript 2 h minus 3 x subscript 1 h squared minus h cubed over denominator h left parenthesis x subscript 1 plus h right parenthesis cubed space x subscript 1 superscript 3 end fraction close parentheses end cell row blank equals cell stack text lim end text with h rightwards arrow 0 below open parentheses fraction numerator negative 3 x subscript 1 superscript 2 minus 3 x subscript 1 h minus h squared over denominator left parenthesis x subscript 1 plus h right parenthesis cubed space x subscript 1 superscript 3 end fraction close parentheses end cell row blank equals cell fraction numerator negative 3 x subscript 1 superscript 2 minus 3 x subscript 1 left parenthesis 0 right parenthesis minus left parenthesis 0 right parenthesis squared over denominator open parentheses x subscript 1 plus 0 close parentheses cubed space x subscript 1 superscript 3 end fraction end cell row blank equals cell fraction numerator negative 3 x subscript 1 superscript 2 over denominator left parenthesis x subscript 1 right parenthesis cubed space x subscript 1 superscript 3 end fraction end cell row blank equals cell fraction numerator negative 3 over denominator x subscript 1 superscript 4 end fraction end cell end table end style$

Dengan demikian, didapat garis singgung pada titik tersebut adalah sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript g s end subscript end cell equals cell A minus B end cell row blank equals cell 3 x subscript 1 superscript 2 minus fraction numerator negative 3 over denominator x subscript 1 superscript 4 end fraction end cell row blank equals cell 3 x subscript 1 superscript 2 plus fraction numerator 3 over denominator x subscript 1 superscript 4 end fraction end cell end table end style$

Jadi, jawaban yang tepat adalah D.