Pertanyaan

Turunan pertama dari fungsi y = tan − 1 x adalah...

Turunan pertama dari fungsi $y = tan^{- 1} x$ adalah...

$fraction numerator 1 over denominator 2 open parentheses 1 plus x squared close parentheses square root of tan to the power of negative 1 end exponent x end root end fraction$
$negative fraction numerator 1 over denominator 2 open parentheses 1 plus x squared close parentheses square root of tan to the power of negative 1 end exponent x end root end fraction$
$fraction numerator 1 over denominator 2 x square root of tan to the power of negative 1 end exponent x end root end fraction$
$negative fraction numerator 1 over denominator 2 x square root of tan to the power of negative 1 end exponent x end root end fraction$
$negative fraction numerator 1 over denominator 2 x squared plus 1 square root of tan to the power of negative 1 end exponent x end root end fraction$

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Ingatlah! Gunakan aturan, Maka, Cari turunan Subtitusi Jadi, turunan pertama adalah . Oleh karena itu, jawaban yang benar adalah A.

Ingatlah!

$table attributes columnalign right center left columnspacing 0px end attributes row cell t a n to the power of negative 1 end exponent x end cell equals cell a r c tan left parenthesis x right parenthesis end cell row cell fraction numerator d over denominator d x end fraction open parentheses square root of t a n to the power of negative 1 end exponent x end root close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses square root of t a n to the power of negative 1 end exponent x end root close parentheses end cell end table$

Gunakan aturan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell square root of u end cell row u equals cell a r c tan left parenthesis x right parenthesis end cell end table$

Maka,

$fraction numerator d over denominator d x end fraction open parentheses square root of arctan open parentheses x close parentheses end root close parentheses equals fraction numerator d over denominator d u end fraction open parentheses square root of u close parentheses fraction numerator d over denominator d x end fraction open parentheses arctan open parentheses x close parentheses close parentheses$

Cari turunan $fraction numerator d over denominator d u end fraction open parentheses square root of u close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell square root of a end cell equals cell a to the power of 1 half end exponent end cell row cell fraction numerator d blank over denominator d x end fraction left parenthesis x to the power of a right parenthesis end cell equals cell a x to the power of a minus 1 end exponent end cell row cell fraction numerator d over denominator d u end fraction open parentheses square root of u close parentheses end cell equals cell fraction numerator d over denominator d u end fraction open parentheses u to the power of 1 half end exponent close parentheses end cell row blank equals cell 1 half u to the power of 1 half minus 1 end exponent end cell row blank equals cell 1 half u to the power of 1 half minus 2 over 2 end exponent end cell row blank equals cell 1 half u to the power of negative 1 half end exponent end cell row blank equals cell 1 half times fraction numerator 1 over denominator square root of u end fraction end cell row blank equals cell fraction numerator 1 times thin space 1 over denominator 2 square root of u end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of u end fraction end cell row blank blank blank end table$

Subtitusi u equals arctan open parentheses x close parentheses

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 over denominator 2 square root of u end fraction fraction numerator d over denominator d x end fraction open parentheses arctan open parentheses x close parentheses close parentheses end cell equals cell fraction numerator 1 over denominator 2 square root of arctan open parentheses x close parentheses end root end fraction fraction numerator d over denominator d x end fraction open parentheses arctan open parentheses x close parentheses close parentheses end cell row blank equals cell fraction numerator 1 over denominator 2 square root of arctan open parentheses x close parentheses end root end fraction times fraction numerator 1 over denominator x squared plus 1 end fraction end cell row blank equals cell fraction numerator 1 times thin space 1 over denominator 2 square root of arctan open parentheses x close parentheses end root open parentheses x squared plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of arctan open parentheses x close parentheses end root open parentheses x squared plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of arctan open parentheses x close parentheses end root open parentheses x squared plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 square root of tan to the power of negative 1 end exponent left parenthesis x right parenthesis end root open parentheses x squared plus 1 close parentheses end fraction end cell end table$

Jadi, turunan pertama y equals square root of tan to the power of negative 1 end exponent x end root adalah $fraction numerator 1 over denominator 2 open parentheses 1 plus x squared close parentheses square root of tan to the power of negative 1 end exponent x end root end fraction$ .

Oleh karena itu, jawaban yang benar adalah A.

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