Pertanyaan

Jika diketahui kurva x 3 − 3 x y 3 = 0 , maka d x d y = .....

Jika diketahui kurva $x^{3} - 3 x y^{3} = 0$ , maka $\frac{d y}{d x} =$ .....

$begin mathsize 14px style fraction numerator 2 x squared square root of x minus square root of y cubed end root over denominator 3 square root of x squared y end root end fraction end style$
$begin mathsize 14px style fraction numerator x squared square root of x minus 2 square root of y cubed end root over denominator 3 square root of x squared y end root end fraction end style$
$begin mathsize 14px style fraction numerator square root of y cubed end root minus 2 x squared square root of x over denominator 3 square root of x squared y end root end fraction end style$
$begin mathsize 14px style fraction numerator 2 square root of y cubed end root minus x squared square root of x over denominator 3 square root of x squared y end root end fraction end style$
$begin mathsize 14px style fraction numerator 2 y squared square root of y minus square root of x cubed end root over denominator 3 square root of x squared y end root end fraction end style$

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Perhatikan bahwa Selanjutnya, perhatikan bahwa Kemudian, didapat bahwa Oleh karena itu, Jadi, jawaban yang tepat adalah A.

Perhatikan bahwa

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x cubed minus 3 square root of x y cubed end root end cell equals 0 row cell fraction numerator d over denominator d x end fraction open parentheses x cubed minus 3 square root of x y cubed end root close parentheses end cell equals 0 row cell fraction numerator d over denominator d x end fraction open parentheses x cubed close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 3 square root of x y cubed end root close parentheses end cell equals 0 row cell fraction numerator d over denominator d x end fraction open parentheses x cubed close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 3 square root of x times square root of y cubed end root close parentheses end cell equals 0 row cell fraction numerator d over denominator d x end fraction open parentheses x cubed close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 3 x to the power of 1 half end exponent times y to the power of 3 over 2 end exponent close parentheses end cell equals 0 end table end style$

Selanjutnya, perhatikan bahwa

$begin mathsize 14px style fraction numerator d over denominator d x end fraction open parentheses x cubed close parentheses equals 3 x squared end style$

Kemudian, didapat bahwa

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses 3 x to the power of 1 half end exponent times y to the power of 3 over 2 end exponent close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses 3 x to the power of 1 half end exponent close parentheses times y to the power of 3 over 2 end exponent plus 3 x to the power of 1 half end exponent times fraction numerator d over denominator d x end fraction open parentheses y to the power of 3 over 2 end exponent close parentheses end cell row blank equals cell 3 times 1 half times x to the power of negative 1 half end exponent times y to the power of 3 over 2 end exponent plus 3 x to the power of 1 half end exponent times 3 over 2 times y to the power of 1 half end exponent times fraction numerator d y over denominator d x end fraction end cell row blank equals cell 3 over 2 times fraction numerator 1 over denominator square root of x end fraction times square root of y cubed end root plus 9 over 2 times square root of x times square root of y times fraction numerator d y over denominator d x end fraction end cell row blank equals cell fraction numerator 3 square root of y cubed end root over denominator 2 square root of x end fraction plus fraction numerator 9 square root of x y end root over denominator 2 end fraction times fraction numerator d y over denominator d x end fraction end cell end table end style$

Oleh karena itu,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d over denominator d x end fraction open parentheses x cubed close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 3 x to the power of 1 half end exponent times y to the power of 3 over 2 end exponent close parentheses end cell equals 0 row cell 3 x squared minus open parentheses fraction numerator 3 square root of y cubed end root over denominator 2 square root of x end fraction plus fraction numerator 9 square root of x y end root over denominator 2 end fraction times fraction numerator d y over denominator d x end fraction close parentheses end cell equals 0 row cell 3 x squared minus fraction numerator 3 square root of y cubed end root over denominator 2 square root of x end fraction minus fraction numerator 9 square root of x y end root over denominator 2 end fraction times fraction numerator d y over denominator d x end fraction end cell equals 0 row cell 3 x squared minus fraction numerator 3 square root of y cubed end root over denominator 2 square root of x end fraction end cell equals cell fraction numerator 9 square root of x y end root over denominator 2 end fraction times fraction numerator d y over denominator d x end fraction end cell row cell fraction numerator 6 x squared square root of x over denominator 2 square root of x end fraction minus fraction numerator 3 square root of y cubed end root over denominator 2 square root of x end fraction end cell equals cell fraction numerator 9 square root of x y end root over denominator 2 end fraction times fraction numerator d y over denominator d x end fraction end cell row cell fraction numerator 6 x squared square root of x minus 3 square root of y cubed end root over denominator 2 square root of x end fraction end cell equals cell fraction numerator 9 square root of x y end root over denominator 2 end fraction times fraction numerator d y over denominator d x end fraction end cell row cell fraction numerator d y over denominator d x end fraction end cell equals cell fraction numerator 6 x squared square root of x minus 3 square root of y cubed end root over denominator 2 square root of x end fraction times fraction numerator 2 over denominator 9 square root of x y end root end fraction end cell row cell fraction numerator d y over denominator d x end fraction end cell equals cell fraction numerator 3 open parentheses 2 x squared square root of x minus square root of y cubed end root close parentheses over denominator 2 square root of x end fraction times fraction numerator 2 over denominator 9 square root of x y end root end fraction end cell row cell fraction numerator d y over denominator d x end fraction end cell equals cell fraction numerator 2 x squared square root of x minus square root of y cubed end root over denominator 3 square root of x squared y end root end fraction end cell end table end style$

Jadi, jawaban yang tepat adalah A.