Pertanyaan

Tentukan turunan pertama fungsi berikut, sederhanakan jika mungkin. 3 . y = tan − 1 x ln x

Tentukan turunan pertama fungsi berikut, sederhanakan jika mungkin.

$3$ . $y = tan^{- 1} x ln x$

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Habis dalam

Klaim

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Pembahasan

Teorema turunan fungsi invers trigonometri. Gunakan rumus turunan perkalian, yaitu: Sehingga diperoleh: dan Kemudian turunannya yaitu: Maka, turunan pertama fungsi adalah .

Teorema turunan fungsi invers trigonometri.

$fraction numerator d over denominator d x end fraction open square brackets tan to the power of negative 1 end exponent space x close square brackets equals fraction numerator 1 over denominator 1 plus x squared end fraction$

Gunakan rumus turunan perkalian, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell u times v end cell row cell y apostrophe end cell equals cell u apostrophe times v plus u times v apostrophe end cell end table

Sehingga diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row u equals cell tan to the power of negative 1 end exponent space x end cell row cell u apostrophe end cell equals cell fraction numerator 1 over denominator 1 plus x squared end fraction end cell end table$

dan

v equals ln space x rightwards arrow v apostrophe equals 1 over x

Kemudian turunannya yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell tan to the power of negative 1 end exponent space x space ln space x end cell row cell y apostrophe end cell equals cell u apostrophe times v plus u times v apostrophe end cell row blank equals cell fraction numerator 1 over denominator 1 plus x squared end fraction times ln space x plus tan to the power of negative 1 end exponent space x times 1 over x end cell row blank equals cell fraction numerator ln space x over denominator 1 plus x squared end fraction plus fraction numerator tan to the power of negative 1 end exponent space x over denominator x end fraction end cell end table$

Maka, turunan pertama fungsi y equals tan to the power of negative 1 end exponent space x space ln space x adalah $y apostrophe equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator ln space x over denominator 1 plus x squared end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator tan to the power of negative 1 end exponent space x over denominator x end fraction end cell end table$ .