Pertanyaan

Tentukan turunan dari fungsi di bawah ini. 2. y = tan 2 ( 2 x − 1 x 2 + x + 2 )

Tentukan turunan dari fungsi di bawah ini.

2. $y = tan^{2} (\frac{x ^{2} + x + 2}{2 x - 1})$

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

turunan adalah .

turunan $y equals tan squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses$ adalah $fraction numerator y apostrophe equals 2 sec squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses open parentheses 2 x squared minus 2 x minus 5 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction$ .

Pembahasan

Ingatlah! Maka, Cari turunan Subtitusi , Cari turunan Turunan Maka, Di dapatkan, Jadi, turunan adalah .

Ingatlah!

$fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction equals fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction$

Maka,

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row f equals cell u squared end cell row cell thin space u end cell equals cell tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses end cell row blank blank blank row cell fraction numerator d over denominator d x end fraction open parentheses tan squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses close parentheses end cell equals cell fraction numerator d over denominator d u end fraction open parentheses u squared close parentheses fraction numerator d over denominator d x end fraction open parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses close parentheses end cell end table end style$

Cari turunan $fraction numerator d over denominator d u end fraction open parentheses u squared close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d blank over denominator d x end fraction open parentheses x to the power of a close parentheses end cell equals cell a x to the power of a minus 1 end exponent end cell row cell fraction numerator d over denominator d u end fraction open parentheses u squared close parentheses end cell equals cell 2 u to the power of 2 minus 1 end exponent end cell row blank equals cell 2 u end cell end table$

Subtitusi $u equals tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses$ ,

$2 u equals 2 tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses$

Cari turunan $fraction numerator d over denominator d x end fraction open parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d f left parenthesis u right parenthesis over denominator d x end fraction end cell equals cell fraction numerator d f over denominator d u end fraction times fraction numerator d u over denominator d x end fraction end cell row f equals cell tan left parenthesis u right parenthesis end cell row u equals cell fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction end cell row cell fraction numerator d over denominator d x end fraction open parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses close parentheses end cell equals cell fraction numerator d over denominator d u end fraction open parentheses tan open parentheses u close parentheses close parentheses fraction numerator d over denominator d x end fraction open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses end cell row blank equals cell sec squared open parentheses u close parentheses fraction numerator d over denominator d x end fraction open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses end cell end table$

Turunan $fraction numerator d over denominator d x end fraction open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses$

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f over g close parentheses to the power of apostrophe end cell equals cell fraction numerator f apostrophe times g minus g apostrophe times f over denominator g squared end fraction end cell row f equals cell x squared plus x plus 2 end cell row g equals cell 2 x plus 1 end cell end table$

Maka,

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator d over denominator d x end fraction open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses end cell row blank equals cell fraction numerator fraction numerator d over denominator d x end fraction open parentheses x squared plus x plus 2 close parentheses open parentheses 2 x minus 1 close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 2 x minus 1 close parentheses open parentheses x squared plus x plus 2 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell row blank blank blank row blank blank cell fraction numerator d over denominator d x end fraction open parentheses x squared plus x plus 2 close parentheses end cell row blank equals cell fraction numerator d over denominator d x end fraction open parentheses x squared close parentheses plus fraction numerator d over denominator d x end fraction open parentheses x close parentheses plus fraction numerator d over denominator d x end fraction open parentheses 2 close parentheses end cell row blank equals cell 2 x plus 1 plus 0 end cell row blank equals cell 2 x plus 1 end cell row blank blank blank row cell fraction numerator d over denominator d x end fraction open parentheses 2 x minus 1 close parentheses end cell equals cell fraction numerator d over denominator d x end fraction open parentheses 2 x close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 1 close parentheses end cell row blank equals cell 2 minus 0 end cell row blank equals 2 row blank blank blank row blank blank cell fraction numerator d over denominator d x end fraction open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses end cell row blank equals cell fraction numerator fraction numerator d over denominator d x end fraction open parentheses x squared plus x plus 2 close parentheses open parentheses 2 x minus 1 close parentheses minus fraction numerator d over denominator d x end fraction open parentheses 2 x minus 1 close parentheses open parentheses x squared plus x plus 2 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 2 x plus 1 close parentheses open parentheses 2 x minus 1 close parentheses minus 2 open parentheses x squared plus x plus 2 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator 4 x squared minus 1 minus 2 open parentheses x squared plus x plus 2 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator 4 x squared minus 1 minus 2 x squared minus 2 x minus 4 over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator 2 x squared minus 2 x minus 5 over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell end table end style$

Di dapatkan,

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank blank row blank blank cell fraction numerator d over denominator d x end fraction open parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses close parentheses end cell row blank equals cell 2 tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses sec squared open parentheses u close parentheses fraction numerator d over denominator d x end fraction open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses end cell row blank equals cell 2 tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses sec squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses open parentheses fraction numerator 2 x squared minus 2 x minus 5 over denominator open parentheses 2 x minus 1 close parentheses squared end fraction close parentheses end cell row blank equals cell 2 tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses fraction numerator sec squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses open parentheses 2 x squared minus 2 x minus 5 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator 2 sec squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses open parentheses 2 x squared minus 2 x minus 5 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction end cell end table$

Jadi, turunan $y equals tan squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses$ adalah $fraction numerator y apostrophe equals 2 sec squared open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses tan open parentheses fraction numerator x squared plus x plus 2 over denominator 2 x minus 1 end fraction close parentheses open parentheses 2 x squared minus 2 x minus 5 close parentheses over denominator open parentheses 2 x minus 1 close parentheses squared end fraction$ .