Pertanyaan

Tentukan turunan pertama fungsi berikut, sederhanakan jika mungkin. 4 . f ( t ) = e s e c − 1 t

Tentukan turunan pertama fungsi berikut, sederhanakan jika mungkin.

$4$ . $f (t) = e^{s e c^{- 1} t}$

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Pembahasan

Teorema turunan fungsi invers trigonometri. Rumus turunan fungsi ekspoensial yaitu: Kemudian diperoleh turunannya yaitu: Maka, turunan pertama fungsi adalah .

Teorema turunan fungsi invers trigonometri.

$fraction numerator d over denominator d x end fraction open square brackets sec to the power of negative 1 end exponent italic space x close square brackets equals fraction numerator 1 over denominator x square root of x squared minus 1 end root end fraction$

Rumus turunan fungsi ekspoensial yaitu:

f open parentheses x close parentheses equals e to the power of g x end exponent rightwards arrow f apostrophe open parentheses x close parentheses equals e to the power of g x end exponent times g apostrophe open parentheses x close parentheses

Kemudian diperoleh turunannya yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses t close parentheses end cell equals cell e to the power of sec to the power of negative 1 end exponent space t end exponent end cell row cell f apostrophe open parentheses t close parentheses end cell equals cell e to the power of sec to the power of negative 1 end exponent space t end exponent times fraction numerator 1 over denominator t times t square root of begin display style t squared over t squared end style minus begin display style 1 over t squared end style end root end fraction end cell row blank equals cell e to the power of sec to the power of negative 1 end exponent space t end exponent times fraction numerator 1 over denominator t squared square root of 1 minus begin display style 1 over t squared end style end root end fraction end cell row blank equals cell fraction numerator e to the power of sec to the power of negative 1 end exponent space t end exponent space open vertical bar t close vertical bar over denominator t squared square root of t squared minus 1 end root end fraction end cell end table$

Maka, turunan pertama fungsi f open parentheses t close parentheses equals e to the power of sec to the power of negative 1 end exponent space t end exponent adalah $f apostrophe open parentheses t close parentheses equals fraction numerator e to the power of sec to the power of negative 1 end exponent space t end exponent space open vertical bar t close vertical bar over denominator t squared square root of t squared minus 1 end root end fraction$ .